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Separation functions and mild topologies

Separation functions and mild topologies 1IntroductionLet MMand NNbe Hausdorff topological spaces, with topologies τM{\tau }_{M}and τN{\tau }_{N}. (Sometimes we may also assume that MMor NNbe metric spaces with distances dM{d}_{M}, or respectively dN{d}_{N}; then τM{\tau }_{M}, or respectively τN{\tau }_{N}, will be the associated topology.)In Section 3, we discuss a topology for the space C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)of continuous maps f:M→Nf:M\to N, which we will call “mild topology.”To define and discuss the properties of the mild topology, we have developed a novel method whereby we define a topology τ\tau on a generic set XXby using a family of “separation functions;” this method is presented in Section 2. Separation functions are somewhat similar to the usual distance function in metric spaces (M,d)\left(M,d), but they have weaker hypotheses (so they can be more manageable in some contexts). Separation functions are used to define pseudo balls that are a global base for a T2 topology τ\tau (this is proven in Theorem 2.3). Under some additional hypotheses, we will define “set separation functions” (similar to set distance functions) in Section 2.5 to prove that the topology is T6. Moreover, under further hypotheses, the topology τ\tau is, in fact, metrizable (Theorems 2.21 and 2.30) – such will be the case for the mild topology. We will also discuss other applications of this separation function method. In Section 2.3, we show that the topology on the Sorgenfrey line (that is not metrizable) can be defined by using a suitable family of separation functions. In Section 2.12, we show how separation functions can be easily defined for topological manifolds starting from the atlas of the manifold. These examples show that the method of separation functions is a promising tool that may have further uses beyond the definition of mild topology.After developing this method of separation functions in Section 2, we then proceed, in Section 3, to apply it to the case X=C0(M;N)X={C}^{0}\left(M;\hspace{0.33em}N)to define the mild topology.But first and foremost, we would like to explain why we may find useful a new topology on C0(M;N){C}^{0}\left(M;\hspace{0.33em}N). For this purpose, in Section 1.1 we recapitulate the definitions of “strong” and “weak” topologies; in Section 1.2, assuming N=RnN={{\mathbb{R}}}^{n}, we relate those to the usual definition of Frechét spaces C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}); then, in Sections 1.3, 1.4 and 1.5, we express some properties of these topologies.1.1Topologies for continuous mapsWe define the “graph” of the function f:M→Nf:M\to Nas graph(f)=def{(x,y)∈M×N:y=f(x)}.\hspace{0.1em}\text{graph}\hspace{0.1em}(f)\mathop{=}\limits^{{\rm{def}}}\left\{\left(x,y)\in M\times N:y=f\left(x)\right\}.We distinguish two fundamental examples of topologies for C0=C0(M;N){C}^{0}={C}^{0}\left(M;\hspace{0.33em}N).Definition 1.1The compact-open topology is generated by sets of the form {f∈C0:f(K)⊆U},\{f\in {C}^{0}:f\left(K)\subseteq U\},where K⊆MK\subseteq Mis compact and U⊆NU\subseteq Nis open. (This collection of sets is a subbase for the topology, but it does not always form a base for a topology.) It is also called the “topology of uniform convergence on compact sets” or the “weak topology” in [2]. We will write CW0(M;N){C}_{W}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.Definition 1.2The graph topology is generated by sets of the form {f∈C0(M;N):graph(f)⊆U},\{f\in {C}^{0}\left(M;\hspace{0.33em}N):\hspace{0.1em}\text{graph}\hspace{0.1em}(f)\subseteq U\},where UUruns through all open sets in M×NM\times N. It is also called “wholly open topology” in [4], “fine” or “Whitney” or “strong topology” in [2]. We will write CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.An equivalent definition of the strong topology can be formulated under additional hypotheses.Proposition 1.3(41.6 in [4], or Chapter 2 Section 4 in [2]) If M is paracompact and (N,dN)\left(N,{d}_{N})is a metric space, then for f∈C0(M,N)f\in {C}^{0}\left(M,N)the sets {g∈C0(M,N):dN(g(x),f(x))<ε(x)∀x∈M}\left\{g\in {C}^{0}\left(M,N):{d}_{N}\left(g\left(x),f\left(x))\lt \varepsilon \left(x)\forall x\in M\right\}form a base of neighborhoods for the graph topology, where ε\varepsilon runs through all positive continuous functions on M.Another way to state this resultExercise 3 in Chapter 2, Section 4 in [2], to be compared with the “counterexample 1.1.8” in [1].is to define the distances dε(f,g)=defsupx∈Mε(x)dN(g(x),f(x));{d}_{\varepsilon }(f,g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\varepsilon \left(x){d}_{N}\left(g\left(x),f\left(x));then the topology generated by all these distances is the graph topology.It is possible to define similar concepts for Cr(M;N){C}^{r}\left(M;\hspace{0.33em}N), the space of rrtimes differentiable maps between two differentiable manifolds M,NM,N. In this case, we do not detail the discussion.The aforementioned topologies are invariant in the sense that the next proposition shows.Proposition 1.4If ΦM:M˜→M{\Phi }_{M}:\tilde{M}\to Mand ΦN:N→N˜{\Phi }_{N}:N\to \tilde{N}are homeomorphisms, then the map f↦ΦN∘f∘ΦMf\mapsto {\Phi }_{N}\circ f\circ {\Phi }_{M}is a homeomorphism between C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)and C0(M˜;N˜){C}^{0}\left(\tilde{M};\hspace{0.33em}\tilde{N}), where the spaces are both endowed either with the “weak” or the “strong” topology.1.2When N=RnN={{\mathbb{R}}}^{n}Let us suppose in this section that NNis the standard Euclidean space Rn{{\mathbb{R}}}^{n}and that MMis a Hausdorff locally compact and second countableThat is, it admits a countable base of open sets.topological space.We recall that any second countable Hausdorff space that is locally compact is paracompact, so Proposition 1.3 applies in the current context; moreover, there exists a countable locally finite covering of open sets, each with compact closure.Definition 1.5Cloc0(M,Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M,{{\mathbb{R}}}^{n})is the Frechét space whose topology is generated by the seminorms [f]K=supx∈K∣f(x)∣,{[f]}_{K}=\mathop{\sup }\limits_{x\in K}| f\left(x)| ,for K⊆MK\subseteq Mcompact. If MMis compact, then it coincides with the usual Banach space C0(M,Rn){C}^{0}\left(M,{{\mathbb{R}}}^{n})associated with the norm ‖f‖=supx∈M∣f(x)∣.\Vert f\Vert =\mathop{\sup }\limits_{x\in M}| f\left(x)| .Proposition 1.6Under the above hypotheses, CW0(M;Rn){C}_{W}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})coincides with Cloc0(M,Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M,{{\mathbb{R}}}^{n}).We can define also another topology.For K⊆MK\subseteq M, compact we define the subset V0,K={f∈C0(M;Rn):supp(f)⊆K}{V}_{0,K}=\{f\in {C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}):\hspace{0.1em}\text{supp}\hspace{0.1em}(f)\subseteq K\}of C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})functions with support in KK. Each such V0,K{V}_{0,K}is a closed subspace of Cloc0(M;Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}). Then Cloc0(M;Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})is a Frechét space with the induced topology.We can then define this topology.Definition 1.7(Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology) The Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology is the strict inductive limitFor the definition of strict inductive limit and its properties, we refer to 17G at page 148 in [3].with respect to the inclusions V0,K→C0(M;Rn){V}_{0,K}\to {C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})for K⊆MK\subseteq Mcompact. A set WWis open in the Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology if, for all K⊆MK\subseteq Mcompact, W∩V0,KW\cap {V}_{0,K}is open in V0,K{V}_{0,K}.1.3Properties of weak and strong topologiesThe weak topology enjoys nice metrization properties.Proposition 1.8Let N be metrizable with a complete metric, and let MMbe locally compact and second countable. Then CW0(M,N){C}_{W}^{0}\left(M,N)has a complete metric.This is proven in Theorem 4.1 in Chapter 2, Section 4 in [2].We ponder on these remarks, taken from [2].Remark 1.9The topological space CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)resulting from the strong topology is the same as CW0(M;N){C}_{W}^{0}\left(M;\hspace{0.33em}N)if MMis compact. If MMis not compact, however, CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)can be an extremely large topology; for example, when M,NM,Nare differentiable finite dimensional manifolds (of positive dimension), then it is not metrizable and in fact it does not have a countable local base at any point, and it has uncountably many connected components.In particular, if MMis not compact, CS0(M;Rn){C}_{S}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})cannot be a topological vector space, but the connected component containing f≡0f\equiv 0coincides with Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}).1.4Proper mapsDefinition 1.10A proper map f:M→Nf:M\to Nis a continuous map such that we have that f−1(K){f}^{-1}\left(K)is compact in MM, for any K⊆NK\subseteq Ncompact.Obviously, if MMis compact then any continuous function is proper. More in general:Lemma 1.11The set of proper maps f:M→Nf:M\to Nis both open and closed in the strong CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)topology.For the proof see, e.g., Section 5.1 in [7] or Theorem 1.5 in Chapter 2, Section 1 in [2].It is easily seen that, in general, the set of proper functions is neither closed nor open in any Clocr{C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{r}topology.Example 1.12Consider functions f:R→Rf:{\mathbb{R}}\to {\mathbb{R}}, and the weak topology; it is easy to show examples of sequences fn→f{f}_{n}\to fsuch that none of the fn{f}_{n}are proper, but ffis, e.g., fn(x)=narctan(x/n){f}_{n}\left(x)=n\arctan \left(x\hspace{0.1em}\text{/}\hspace{0.1em}n)and f(x)=xf\left(x)=x;fn{f}_{n}are all proper but ffis not, e.g., fn(x)=1nx2{f}_{n}\left(x)=\frac{1}{n}{x}^{2}and f(x)=0f\left(x)=0.The aforementioned examples hold also in Cloc∞{C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\infty }, where convergence is defined as “local uniform convergence of all derivatives.”1.5Drawback of strong topologiesGiven the above discussion, it would seem that, when dealing with proper maps, it would be better to use a strong topology. Strong topologies have drawbacks as well.In particular, consider the case of maps f:M→Rnf:M\to {{\mathbb{R}}}^{n}, let C0=C0(M;Rn){C}^{0}={C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}); we have some natural actions: translations, given by the action (v,f)∈Rn×C0↦f+v∈C0;\left(v,f)\in {{\mathbb{R}}}^{n}\times {C}^{0}\mapsto f+v\in {C}^{0};rotations (A,f)∈SO(n)×C0↦Af∈C0;\left(A,f)\in {\rm{\text{SO}}}\left(n)\times {C}^{0}\mapsto Af\in {C}^{0};rescalings (λ,f)∈I×C0↦λf∈C0,\left(\lambda ,f)\in I\times {C}^{0}\mapsto \lambda f\in {C}^{0},for I=(0,∞)⊂RI=\left(0,\infty )\subset {\mathbb{R}}and, in general, affine transformations F,f↦Ff,F,f\mapsto Ff,where Fy=Ay+vFy=Ay+vis given by A∈GL(Rn)A\in \hspace{0.1em}\text{GL}\hspace{0.1em}\left({{\mathbb{R}}}^{n}), v∈Rnv\in {{\mathbb{R}}}^{n}.These actions are continuous if C0{C}^{0}is endowed with the weak topology, but it may fail to be continuous if C0{C}^{0}is endowed with the strong topology.There is another drawback (that we already remarked): when N=RnN={{\mathbb{R}}}^{n}, then C0{C}^{0}with the strong topology may fail to be a topological vector space.We will show in Section 3 that the mild topology shares some good properties valid for the weak and the strong topology.2Topology by separation functionsIn this section, given a generic set XX, we will use “separation functions” to define a topology τ\tau on XX; we will then study the properties of this topology.Definition 2.1A family d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}of real positive functions dx(y):X→[0,∞],{d}_{x}(y):X\to \left[0,\infty ],one for each x∈Xx\in X, is a family of separation functions when dx(y)=0{d}_{x}(y)=0iff x=yx=y;given y∈Xy\in Xand α,β∈R\alpha ,\beta \in {\mathbb{R}}with 0<β<α0\lt \beta \lt \alpha there exists a function (2.1)ρd=ρd(y,α,β)>0{\rho }_{d}={\rho }_{d}(y,\alpha ,\beta )\gt 0(called “modulus”) such that, for all x,z∈Xx,z\in X, (2.2)dx(y)≤β∧dy(z)<ρd⇒dx(z)<α,{d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,(2.3)dx(y)≥α∧dy(z)<ρd⇒dx(z)>β.{d}_{x}(y)\ge \alpha \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\gt \beta .This condition (2.2) will be called “pseudo triangle inequality,” while condition (2.3) will be called “pseudo reverse triangle inequality” (Figure 1).Figure 1Representation of triangle inequalities.These are written as dx(y){d}_{x}(y)and not d(x,y)d\left(x,y)to remark that they do not satisfy the axioms of “distances”: they are not required to be symmetric and do not satisfy the standard triangle inequality.Remark 2.2An “asymmetric distance” (a.k.a. “quasi metric”) is a function b(x,y):X2→[0,+∞]b\left(x,y):{X}^{2}\to \left[0,+\infty ]that satisfies the separation property “b(x,y)=0⇔x=yb\left(x,y)=0\hspace{0.33em}\iff \hspace{0.33em}x=y” and the standard triangle inequality, but it may fail to be symmetric. See [5,6] and references therein. An “asymmetric distance” immediately provides a family of separation functions dx(y)=b(x,y){d}_{x}(y)=b\left(x,y)with ρd(y,α,β)=α−β{\rho }_{d}(y,\alpha ,\beta )=\alpha -\beta .Theorem 2.3Given a set X with a family of separation functions, then we can define “pseudo balls” B(x,ε)={y∈X:dx(y)<ε};B\left(x,\varepsilon )=\{y\in X:{d}_{x}(y)\lt \varepsilon \};these are then a global base for a T2 topology τ\tau , and each B(x,ε)B\left(x,\varepsilon )is an open neighborhood of xxin (X,τ)\left(X,\tau ).ProofIndeed the pseudo triangle inequality (2.2) shows that if y∈B(x,α)y\in B\left(x,\alpha )and β=dx(y)\beta ={d}_{x}(y)and ρd=ρd(y,α,β){\rho }_{d}={\rho }_{d}(y,\alpha ,\beta ), then B(y,ρd)⊆B(x,α).B(y,{\rho }_{d})\subseteq B\left(x,\alpha ).The pseudo reverse triangle inequality (2.3) shows that if dx(y)=α>0{d}_{x}(y)=\alpha \gt 0then □B(x,β)∩B(y,ρd)=∅.B\left(x,\beta )\cap B(y,{\rho }_{d})=\varnothing .Proposition 2.4Each dx(y){d}_{x}(y), for fixed x, is continuous on (X,τ)\left(X,\tau ).ProofTheorem 2.3 readily implies that dx(y){d}_{x}(y)is upper semicontinuous: indeed we already know that B(x,ε)={y∈X:dx(y)<ε}B\left(x,\varepsilon )=\{y\in X:{d}_{x}(y)\lt \varepsilon \}is open. Let V={z∈X:dx(z)>ε};V=\left\{z\in X:{d}_{x}\left(z)\gt \varepsilon \right\};we want to prove that it is open. Let y∈Vy\in V, let α=dx(y)>ε\alpha ={d}_{x}(y)\gt \varepsilon , and let ε<β<α\varepsilon \lt \beta \lt \alpha : the pseudo reverse triangle inequality (2.3) shows that □B(y,ρd)⊆V.B(y,{\rho }_{d})\subseteq V.It is interesting to note that separation functions have a form of stability that makes them more manageable than distance functions.Proposition 2.5Suppose that dx(y){d}_{x}(y)is a separation function and φ:[0,∞]→[0,∞]\varphi :\left[0,\infty ]\to \left[0,\infty ]is in the Θ\Theta class (defined in Definition 2.11); let bx(y)=φ(dx(y)),{b}_{x}(y)=\varphi \left({d}_{x}(y)),then bx(y){b}_{x}(y)is a separation function.Remark 2.6A similar proposition holds for distances when φ\varphi is also subadditive. But dx(y)=∣x−y∣2{d}_{x}(y)=| x-y{| }^{2}is a separation function on R{\mathbb{R}}, and it is not a distance.Remark 2.7Note that we did not assume validity of this statement.(2.4)“Given x∈X and α>0 for any 0<β<α there exists ρ>0 such thatdx(y)<ρ∧dy(z)≤β⇒dx(z)<α.”\begin{array}{l}\hspace{0.1em}\text{&#x201C;Given\hspace{0.5em}}x\in X\text{\hspace{0.5em}and\hspace{0.5em}}\alpha \gt 0\text{\hspace{0.5em}for any\hspace{0.5em}}0\lt \beta \lt \alpha \text{\hspace{0.5em}there\hspace{0.5em}exists\hspace{0.5em}}\rho \gt 0\text{\hspace{0.5em}such that}\hspace{0.1em}\\ {d}_{x}(y)\lt \rho \wedge {d}_{y}\left(z)\le \beta \Rightarrow {d}_{x}\left(z)\lt \alpha \hspace{0.1em}\text{.&#x201D;}\end{array}\hspace{8em}(and similarly for a “reverse” version).This raises an (yet) unanswered question. Let us define d˜y(x)=dx(y):{\tilde{d}}_{y}\left(x)={d}_{x}(y):under which conditions d˜y(x){\tilde{d}}_{y}\left(x)is a separation function?2.1On the modulusWe recall that the function ρd{\rho }_{d}defined in equation (2.1) is called “modulus.” We now prove that, given a family of separation functions, there exists a maximum modulus.Proposition 2.8Having fixed a family of separation functions, there exists a maximum modulus ρˆd{\hat{\rho }}_{d}, which can be explicitly defined as follows (for y∈Xy\in Xand 0<β<α0\lt \beta \lt \alpha ): ρˆd(y,α,β)=defmax{r∈[0,∞]:∀x,z∈X,(dx(y)≤β∧dy(z)<r⇒dx(z)<α)∧(dx(y)≥α∧dy(z)<r⇒dx(z)>β)}.{\hat{\rho }}_{d}(y,\alpha ,\beta )\mathop{=}\limits^{{\rm{def}}}\max \{r\in \left[0,\infty ]:\forall x,z\in X,\left({d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\lt r\Rightarrow {d}_{x}\left(z)\lt \alpha )\wedge ({d}_{x}(y)\ge \alpha \wedge {d}_{y}\left(z)\lt r\Rightarrow {d}_{x}\left(z)\gt \beta )\}.ProofThe fact that ρˆd{\hat{\rho }}_{d}is a “modulus” is obvious from the definition, since the set on right-hand side (RHS) encodes the “pseudo triangle inequality” (2.2) and the “pseudo reverse triangle inequality” (2.3).We prove that the formula defining ρˆd{\hat{\rho }}_{d}is correct, i.e., that the set on RHS has positive maximum; to this end, we rewrite it in this form ρˆd(y,α,β)=defmaxA(y,α,β),A(y,α,β)=⋂x,z∈XP(y,α,β,x,z)∩R(y,α,β,x,z),P(y,α,β,x,z)=def{r∈[0,∞]:(dx(y)>β∨dy(z)≥r∨dx(z)<α)},R(y,α,β,x,z)=def{r∈[0,∞]:(dx(y)<α∨dy(z)≥r∨dx(z)>β)};\begin{array}{rcl}{\hat{\rho }}_{d}(y,\alpha ,\beta )& \mathop{=}\limits^{{\rm{def}}}& \max {A}_{(y,\alpha ,\beta )},\\ {A}_{(y,\alpha ,\beta )}& =& \bigcap _{x,z\in X}{P}_{(y,\alpha ,\beta ,x,z)}\cap {R}_{(y,\alpha ,\beta ,x,z)},\\ {P}_{(y,\alpha ,\beta ,x,z)}& \mathop{=}\limits^{{\rm{def}}}& \{r\in \left[0,\infty ]:\left({d}_{x}(y)\gt \beta \vee {d}_{y}\left(z)\ge r\vee {d}_{x}\left(z)\lt \alpha )\},\\ {R}_{(y,\alpha ,\beta ,x,z)}& \mathop{=}\limits^{{\rm{def}}}& \{r\in \left[0,\infty ]:({d}_{x}(y)\lt \alpha \vee {d}_{y}\left(z)\ge r\vee {d}_{x}\left(z)\gt \beta )\};\end{array}then we express the last two terms as P(y,α,β,x,z)=⋂dx(y)≤β∧dx(z)≥α[0,dy(z)],R(y,α,β,x,z)=⋂dx(y)≥α∧dx(z)≤β[0,dy(z)];\begin{array}{rcl}{P}_{(y,\alpha ,\beta ,x,z)}& =& \bigcap _{{d}_{x}(y)\le \beta \wedge {d}_{x}\left(z)\ge \alpha }\left[0,{d}_{y}\left(z)],\\ {R}_{(y,\alpha ,\beta ,x,z)}& =& \bigcap _{{d}_{x}(y)\ge \alpha \wedge {d}_{x}\left(z)\le \beta }\left[0,{d}_{y}\left(z)];\end{array}eventually we write A(y,α,β)=[0,∞]∩(⋂x,z∈X,dx(y)≤β∧dx(z)≥α[0,dy(z)])∩(⋂x,z∈X,dx(y)≥α∧dx(z)≤β[0,dy(z)]).{A}_{(y,\alpha ,\beta )}=\left[0,\infty ]\cap \left(\bigcap _{x,z\in X,{d}_{x}(y)\le \beta \wedge {d}_{x}\left(z)\ge \alpha }\left[0,{d}_{y}\left(z)]\right)\cap \left(\bigcap _{x,z\in X,{d}_{x}(y)\ge \alpha \wedge {d}_{x}\left(z)\le \beta }\left[0,{d}_{y}\left(z)]\right).This latter is an intersection of closed intervals starting from zero (included), hence A(y,α,β){A}_{(y,\alpha ,\beta )}is a closed interval of the form [0,ρ]\left[0,\rho ]; moreover, each interval in the RHS contains ρd(y,α,β){\rho }_{d}(y,\alpha ,\beta ), so the maximum is positive.□Remark 2.9It is clear by the aforementioned formulas that ρˆd{\hat{\rho }}_{d}is weakly increasing in α\alpha and weakly decreasing in β\beta ; so we may assume this in Definition 2.1, with no loss of generality.If we add strict monotonicity and continuity, we obtain an interesting proposition.Proposition 2.10Consider the “pseudo triangle inequality” equation (2.2) and these three additional conditions: ∀x,y,z∈X,∀α,β\forall x,y,z\in X,\forall \alpha ,\beta with 0<β<α0\lt \beta \lt \alpha , (2.5)dx(y)≤β∧dy(z)≤ρd⇒dx(z)≤α,{d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\le {\rho }_{d}\Rightarrow {d}_{x}\left(z)\le \alpha ,(2.6)dx(y)<β∧dy(z)≤ρd⇒dx(z)<α,{d}_{x}(y)\lt \beta \wedge {d}_{y}\left(z)\le {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,(2.7)dx(y)<β∧dy(z)<ρd⇒dx(z)<α,{d}_{x}(y)\lt \beta \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,where again ρd=ρd(y,α,β){\rho }_{d}={\rho }_{d}(y,\alpha ,\beta )is the same function (2.1) as used in equation (2.2).In the above four formulas (2.2), (2.5), (2.6), and (2.7), we have alternated strict and loose inequalities.Suppose that the function ρd(y,α,β){\rho }_{d}(y,\alpha ,\beta ), for fixed yy, is continuous in α,β\alpha ,\beta , strictly increasing in α\alpha and strictly decreasing in β\beta : then the four conditions (2.2), (2.5), (2.6), and (2.7) are equivalent.(The proof is in Section A.1 in page 19)This happens for distances, where ρd=α−β{\rho }_{d}=\alpha -\beta ; and it happens in Section 3.A similar statement holds for the “pseudo reverse triangle inequality” (2.3): we skip it for brevity.2.2Fundamental familyIn the aforementioned sections, we can assume that there are many families of separation functions di=(dx,i)x∈X{d}_{i}={\left({d}_{x,i})}_{x\in X}, for i∈Ii\in Ia family of indexes (not depending on xx); this is analogous to the framework in locally convex topological vector spaces, where we have multiple seminorms that are used to define multiple balls centered at zero (and then translated to all other points).But, in the following considerations, we will assume for simplicity that, for each x∈Xx\in X, there is only one separation function. So we have a fundamental family of separation functions d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}. Then the topology satisfies the first countability axiom.2.3The Sorgenfrey lineThe Sorgenfrey line is the set X=RX={\mathbb{R}}with the topology generated by all the half-open intervals [a,b)\left[a,b), where a,b∈R,a<ba,b\in {\mathbb{R}},a\lt b; this family is a global base for the topology. See example 51 in [10].We can define (2.8)b(x,y)=y−xy≥x+∞y<x,b\left(x,y)=\left\{\begin{array}{ll}y-x\hspace{1.0em}& y\ge x\\ +\infty \hspace{1.0em}& y\lt x,\end{array}\right.this is an “asymmetric distance” that generates the above topology on R{\mathbb{R}}; by Remark 2.2 dx(y)=b(x,y){d}_{x}(y)=b\left(x,y)is a fundamental family of separation functions, with ρd(y,α,β)=α−β{\rho }_{d}(y,\alpha ,\beta )=\alpha -\beta .The Sorgenfrey line is a T6 space (a perfectly normal Hausdorff space); it is first countable and separable, but not second countable, so it is not metrizable.This suggests that we need some extra hypotheses to obtain better properties.2.4Pseudo symmetryWe define a convenient class.Definition 2.11(Class Θ\Theta ) We define the class Θ\Theta of functions θ:[0,∞]→[0,∞]\theta :\left[0,\infty ]\to \left[0,\infty ]that are continuous, have θ(0)=0\theta \left(0)=0, and are strictly increasing where they are finite. For such a θ\theta we agree that θ−1∈Θ{\theta }^{-1}\in \Theta is so defined (with a slight abuse of notation) θ−1(s)=defsup{t∈[0,∞):θ(t)<s}.{\theta }^{-1}\left(s)\mathop{=}\limits^{{\rm{def}}}\sup \left\{t\in \left[0,\infty ):\theta \left(t)\lt s\right\}.Equivalently, if D=sup0≤s<∞θ(s),D=\mathop{\sup }\limits_{0\le s\lt \infty }\theta \left(s),then θ−1(s){\theta }^{-1}\left(s)is the usual inverse for s<Ds\lt D, otherwise θ−1(s)=+∞{\theta }^{-1}\left(s)=+\infty Indeed any s≥Ds\ge Dcannot be equal to θ(t)\theta \left(t)for t<∞t\lt \infty , since θ\theta is strictly increasing.. This implies that t=θ−1(s)⇔s=θ(t),t={\theta }^{-1}\left(s)\hspace{0.33em}\iff \hspace{0.33em}s=\theta \left(t),whenever s,t<∞s,t\lt \infty .Remark 2.12This is just a convenient choice, which allows us to simplify notation and analysis. For example, the “tangent” function can be represented in Θ\Theta by defining it as θ(t)=tan(t)s<π/2+∞s≥π/2,\theta \left(t)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}\tan \left(t)\hspace{1.0em}& s\lt \pi \hspace{0.1em}\text{/}\hspace{0.1em}2\\ +\infty \hspace{1.0em}& s\ge \pi \hspace{0.1em}\text{/}\hspace{0.1em}2,\end{array}\right.then its “inverse” is just θ−1(s)=arctan(s){\theta }^{-1}\left(s)=\arctan \left(s), with θ−1(+∞)=π/2{\theta }^{-1}\left(+\infty )=\pi \hspace{0.1em}\text{/}\hspace{0.1em}2.Definition 2.13Let us assume that associated with XXwe have a fundamental family (dx)x∈X{\left({d}_{x})}_{x\in X}of separation functions. We will say that this family is pseudo symmetric if there exists a function θd∈Θ{\theta }_{d}\in \Theta such that (2.9)dy(x)≤θd(dx(y)).{d}_{y}\left(x)\le {\theta }_{d}({d}_{x}(y)).Remark 2.14“Pseudo symmetry” is useful because it tells us that the topology is equivalently generated by the inverse balls B˜(x,ε)={y∈X:dy(x)<ε}.\tilde{B}\left(x,\varepsilon )=\{y\in X:{d}_{y}\left(x)\lt \varepsilon \}.This tells us that limn→∞xn=x⇔dx(xn)→n0⇔dxn(x)→n0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{x}_{n}=x\hspace{0.33em}\iff \hspace{0.33em}{d}_{x}\left({x}_{n}){\to }_{n}0\hspace{0.33em}\iff \hspace{0.33em}{d}_{{x}_{n}}\left(x){\to }_{n}0.Compare Definition 3.4 in [5].2.5Set separation functionDefinition 2.15Mimicking the definition in metric space, we define, for A⊆XA\subseteq X, the set separation function(2.10)dA(y)=definfx∈Adx(y).{d}_{A}(y)\mathop{=}\limits^{{\rm{def}}}\mathop{\inf }\limits_{x\in A}{d}_{x}(y).Lemma 2.16dA(y){d}_{A}(y)is continuous.ProofWe know that dA(y){d}_{A}(y)is upper semicontinuous, so we need to prove that it is lower semicontinuous, that is, that V={y∈X:dA(y)>ε}V=\{y\in X:{d}_{A}(y)\gt \varepsilon \}is open, for ε≥0\varepsilon \ge 0. Let y∈Vy\in Vand let α=dA(y)>ε\alpha ={d}_{A}(y)\gt \varepsilon ; let then ε<β<α\varepsilon \lt \beta \lt \alpha and let r=ρd(y,α,β)r={\rho }_{d}(y,\alpha ,\beta ), where ρd{\rho }_{d}was defined in the pseudo reverse triangle inequality equation (2.3). We prove that B(y,r)⊆V;B(y,r)\subseteq V;indeed for any x∈Ax\in Awe have dx(y)≥α{d}_{x}(y)\ge \alpha : so, if dy(z)<r{d}_{y}\left(z)\lt r, then dx(z)>β{d}_{x}\left(z)\gt \beta and hence dA(z)≥β{d}_{A}\left(z)\ge \beta .□Remark 2.17Since {y∈X:dA(y)=0}⊇A\{y\in X:{d}_{A}(y)=0\}\supseteq Aand the left-hand side (LHS) is closed, in general we have {y∈X:dA(y)=0}⊇A¯.\{y\in X:{d}_{A}(y)=0\}\supseteq \overline{A}.To obtain equality we add pseudo symmetry.Hypothesis 2.18From here on we assume that we have a fundamental family (dx)x∈X{\left({d}_{x})}_{x\in X}of separation functions that is pseudo symmetric.Lemma 2.19Assuming that Hypothesis 2.18 holds then {y∈X:dA(y)=0}=A¯.\{y\in X:{d}_{A}(y)=0\}=\overline{A}.ProofWe need to prove that {y∈X:dA(y)=0}⊆A¯.\{y\in X:{d}_{A}(y)=0\}\subseteq \overline{A}.Let then dA(y)=0{d}_{A}(y)=0, which means that there is a sequence (xn)n⊆A{\left({x}_{n})}_{n}\subseteq Asuch that dxn(y)→n0{d}_{{x}_{n}}(y){\to }_{n}0. Then by pseudo symmetry we obtain y∈Ay\in A.□Corollary 2.20In particular, when Hypothesis 2.18 holds the topological space is T6, a.k.a. a perfectly normal Hausdorff space.2.6MetrizationWe have then a first metrization theorem, following the Urysohn metrization theorem.Theorem 2.21Assuming that Hypothesis 2.18 holds and the topological space (X,τ)\left(X,\tau )is second countable, then it is metrizable.ProofIndeed by Corollary 2.20 the space is T6.□2.7Diameter of pseudo ballsDefinition 2.22Let diam(A)=defsupx,y∈Adx(y){\rm{diam}}\left(A)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x,y\in A}{d}_{x}(y)be the diameter of a set A⊆XA\subseteq X.Lemma 2.23If Hypothesis 2.18 holds, then limε→0diam(B(y,ε))=0.\mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\rm{diam}}\left(B(y,\varepsilon ))=0.If ρd{\rho }_{d}does not depend on yy, then the limit above is uniform in yy.ProofThe map ε↦diam(B(y,ε))\varepsilon \mapsto {\rm{diam}}\left(B(y,\varepsilon ))is monotonic. Let us fix α>0\alpha \gt 0and β=α/2\beta =\alpha \hspace{0.1em}\text{/}\hspace{0.1em}2; we will use the pseudo triangle inequality that is equation (2.2). There is ε>0\varepsilon \gt 0small enough such that θd(ε)<β{\theta }_{d}\left(\varepsilon )\lt \beta and ε<ρd(y,α,β)\varepsilon \lt {\rho }_{d}(y,\alpha ,\beta ). Choose any x,z∈B(y,ε)x,z\in B(y,\varepsilon ). By pseudo symmetry x∈B(y,ε)⇔dy(x)<ε⇒dx(y)≤θd(ε)<β,x\in B(y,\varepsilon )\hspace{0.33em}\iff \hspace{0.33em}{d}_{y}\left(x)\lt \varepsilon \Rightarrow {d}_{x}(y)\le {\theta }_{d}\left(\varepsilon )\lt \beta ,so by (2.2) dx(z)<α{d}_{x}\left(z)\lt \alpha .□2.8Uniform modulus“Uniform modulus” is the case when the modulus ρd{\rho }_{d}does not depend on yy. We saw in Lemma 2.23 that in this case we obtain stronger results. In the following, we discuss further results that use uniformity and pseudo symmetry.Hypothesis 2.24From here on we assume that we have a fundamental family d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}of separation functions that is pseudo symmetric, and where the modulus ρd{\rho }_{d}does not depend on yy.We recall that the topology τ\tau was defined in Theorem 2.3.Lemma 2.25Assuming that Hypothesis 2.24 holds, then the topological space (X,τ)\left(X,\tau )is separable if and only if it is second countable.ProofOne implication is well known.We recall that the family of B(x,α)B\left(x,\alpha ), for x∈Xx\in Xand α>0\alpha \gt 0, is a global base for the topology τ\tau , see Theorem 2.3. Let (xn)n∈N{\left({x}_{n})}_{n\in {\mathbb{N}}}be a dense subset of XX, we prove that B(xn,1/m)B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m)is a global base, for n,m≥1n,m\ge 1integers; to this end for any B(x,α)B\left(x,\alpha )we will prove that there are n,mn,msuch that x∈B(xn,1/m)⊆B(x,α).x\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m)\subseteq B\left(x,\alpha ).Indeed, we choose mmsuch that (2.11)diamB(y,1/m)<α{\rm{diam}}B(y,1\hspace{0.1em}\text{/}\hspace{0.1em}m)\lt \alpha for any yy. Then, by pseudo symmetry, there is an nnsuch that dxn(x)<1/m{d}_{{x}_{n}}\left(x)\lt 1\hspace{0.1em}\text{/}\hspace{0.1em}m: this implies that x∈B(xn,1/m)x\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m). Then, by (2.11), for any z∈B(xn,1/m)z\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m), we have dx(z)<α{d}_{x}\left(z)\lt \alpha , so we conclude that z∈B(x,α)z\in B\left(x,\alpha ).□2.9Forward and reverseProposition 2.26If Hypothesis 2.24 holds, then the “pseudo triangle inequality” (2.2) and the “pseudo reverse triangle inequality” (2.3) are equivalent.ProofWe prove that if (2.2) holds then (2.3) holds. (The other implication is similar.) Consider the formula (2.12)dx(z)≥α∧dz(y)<ρ˜d⇒dx(y)>β{d}_{x}\left(z)\ge \alpha \wedge {d}_{z}(y)\lt {\tilde{\rho }}_{d}\Rightarrow {d}_{x}(y)\gt \beta that is equivalent to (2.3), since it was obtained by switching y,zy,zand replacing the modulus. We prove that for 0<β<α0\lt \beta \lt \alpha there exists ρ˜d=ρ˜d(α,β)>0{\tilde{\rho }}_{d}={\tilde{\rho }}_{d}\left(\alpha ,\beta )\gt 0such that ∀x,y,z∈X\forall x,y,z\in X, (2.12) holds.The condition (2.2) equivalently tells us that dx(y)>β∨dy(z)≥ρd∨dx(z)<α.{d}_{x}(y)\gt \beta \vee {d}_{y}\left(z)\ge {\rho }_{d}\vee {d}_{x}\left(z)\lt \alpha .If dx(z)≥α∧dz(y)<θ−1(ρd),{d}_{x}\left(z)\ge \alpha \wedge {d}_{z}(y)\lt {\theta }^{-1}\left({\rho }_{d}),then (by pseudo symmetry (2.9)) dy(z)≤θd(dz(y))<ρd;{d}_{y}\left(z)\le {\theta }_{d}({d}_{z}(y))\lt {\rho }_{d};hence dx(y)>β{d}_{x}(y)\gt \beta . We then define ρ˜d=θd−1∘ρd{\tilde{\rho }}_{d}={\theta }_{d}^{-1}\circ {\rho }_{d}with the convention in Definition 2.11.□2.10Delta complementWe extract a lemma from the proof of the following Theorem 2.29.Definition 2.27We assume Hypothesis 2.18. Given C⊂XC\subset Xclosed and δ>0\delta \gt 0we define Fδ(C)={x∈X:dC(x)>δ};{F}_{\delta }\left(C)=\left\{x\in X:{d}_{C}\left(x)\gt \delta \right\};we call Fδ(C){F}_{\delta }\left(C)the “delta-complemented set” since Fδ(C)∩C=∅,{F}_{\delta }\left(C)\cap C=\varnothing ,Fδ(C){F}_{\delta }\left(C)is open, and ⋃δ>0Fδ(C)=X⧹C.\bigcup _{\delta \gt 0}{F}_{\delta }\left(C)=X\setminus C.(All this follows from Lemmas 2.16 and 2.19; Figure 2.) Note also that for 0<s<t0\lt s\lt twe have Fs(C)⊇Ft(C),{F}_{s}\left(C)\supseteq {F}_{t}\left(C),while for C1⊆C2{C}_{1}\subseteq {C}_{2}closed sets we have (2.13)Fs(C1)⊇Fs(C2).{F}_{s}\left({C}_{1})\supseteq {F}_{s}\left({C}_{2}).Figure 2δ\delta complement.Lemma 2.28We assume Hypothesis 2.24. Let now s,t∈Rs,t\in {\mathbb{R}}with 0<s<t0\lt s\lt t: there exists an ε>0\varepsilon \gt 0such that, for any w∈Xw\in X, for any C⊆XC\subseteq Xclosed, if B(w,ε)∩Ft(C)≠∅,B\left(w,\varepsilon )\cap {F}_{t}\left(C)\ne \varnothing ,then B(w,ε)⊆Fs(C).B\left(w,\varepsilon )\subseteq {F}_{s}\left(C).ProofLet 0<s<s˜<t0\lt s\lt \tilde{s}\lt t. By the pseudo reverse triangle inequality (2.3), let ρ>0\rho \gt 0be small enough so that dx(z)≥t∧dz(y)<ρ⇒dx(y)>s˜.{d}_{x}\left(z)\ge t\wedge {d}_{z}(y)\lt \rho \Rightarrow {d}_{x}(y)\gt \tilde{s}.Then choose ε>0\varepsilon \gt 0small enough so that, by Lemma 2.23, we have ∀w∈X,∀z,y∈B(w,ε)⇒dz(y)<ρ.\forall w\in X,\forall z,y\in B\left(w,\varepsilon )\Rightarrow {d}_{z}(y)\lt \rho .Let us now suppose that z∈B(w,ε)∩Ft(C).z\in B\left(w,\varepsilon )\cap {F}_{t}\left(C).For all x∈Cx\in C, we have dx(z)>t{d}_{x}\left(z)\gt t. Consider any y∈B(w,ε)y\in B\left(w,\varepsilon ), then we have dz(y)<ρ{d}_{z}(y)\lt \rho . We conclude that dx(y)>s˜{d}_{x}(y)\gt \tilde{s}, hence infx∈Cdx(y)≥s˜{\inf }_{x\in C}{d}_{x}(y)\ge \tilde{s}. We have so proved that y∈Fs(A).y\in {F}_{s}\left(A).□2.11MetrizabilityTheorem 2.29We assume Hypothesis 2.24. If A{\mathcal{A}}is an open covering of X, then there exists ℰ{\mathcal{ {\mathcal E} }}an open covering of X that is countably locally finite, and ℰ{\mathcal{ {\mathcal E} }}is a refinement of A{\mathcal{A}}.(The proof is in Section A.2 on page 91.)Theorem 2.30If we assume Hypothesis 2.24, then the topological space (X,τ)\left(X,\tau )is metrizable.ProofThis follows from the above theorem, Corollary 2.20, and the Nagata-Smirnov metrization theorem (Sections 6-2 and 6-3 in [8]).□2.12Example: topological manifoldsAs an example, we propose this construction. In this section, we consider a topological manifold (X,τ)\left(X,\tau )that is Hausdorff, second countable, and locally Euclidean with dimension mm. (See §36 in [8] for further details.) Then (X,τ)\left(X,\tau )is paracompact and σ\sigma -compact. So there exists an atlas of homeomorphisms φi:Vi→Rm{\varphi }_{i}:{V}_{i}\to {{\mathbb{R}}}^{m}satisfying these additional conditions: For this atlas we have that the index family is I=NI={\mathbb{N}}, or IIis finite;For each i∈Ii\in Iwe have that Vi⊆X{V}_{i}\subseteq Xis open with compact closure V¯i{\overline{V}}_{i};Moreover, (Vi)i∈I{\left({V}_{i})}_{i\in I}is a locally finite open cover of XX.Theorem 2.31For x,y∈Xx,y\in Xwe can define(2.14)dx(y)=min{∣φi(x)−φi(y)∣Rm:i∈I∧x∈Vi∧y∈Vi}.{d}_{x}(y)=\min \left\{| {\varphi }_{i}\left(x)-{\varphi }_{i}(y){| }_{{{\mathbb{R}}}^{m}}:i\in I\wedge x\in {V}_{i}\wedge y\in {V}_{i}\right\}.Note that the set on the RHS is finite; if it is empty, then we set dx(y)=+∞{d}_{x}(y)=+\infty . These functions dx(y){d}_{x}(y)are continuous (jointly in x,yx,y) and are a fundamental family of separation functions that generates the topology τ\tau of XX.(Although the above result seems to be intuitive, the proof is surprisingly long and intricate, so it was moved to Section A.3.)Interestingly, the modulus ρd{\rho }_{d}associated with the family dx(y){d}_{x}(y)can be defined to satisfy the requirements of Proposition 2.10.Note that dx(y)=dy(x){d}_{x}(y)={d}_{y}\left(x), so this family satisfies Hypothesis 2.18. Consequently, the set separation function dA(y){d}_{A}(y)satisfies Lemmas 2.16 and 2.19: this explicitly proves that the space is T6. So Theorems 2.31 and 2.21 are an alternative way to prove that a manifold (X,τ)\left(X,\tau )is metrizable when it satisfies the hypotheses listed at the beginning of this section.Example 2.32In general, the function dx(y){d}_{x}(y)defined above in (2.14) does not satisfy the triangle inequality. Consider this example of a manifold covered with two charts, where x∈V1⧹V2,y∈V1∩V2,z∈V2⧹V1x\in {V}_{1}\setminus {V}_{2},\hspace{1.0em}y\in {V}_{1}\cap {V}_{2},\hspace{1.0em}z\in {V}_{2}\setminus {V}_{1}and dx(y)<∞,dy(z)<∞,dx(z)=∞.{d}_{x}(y)\lt \infty ,\hspace{1.0em}{d}_{y}\left(z)\lt \infty ,\hspace{1.0em}{d}_{x}\left(z)=\infty .Example 2.33Consider X=RX={\mathbb{R}}and cover it with charts having Vn=(n−1,n+1){V}_{n}=\left(n-1,n+1)and φn(x)=1max{1,∣n∣}ψ(x−n),ψ(x)=x(1−x2){\varphi }_{n}\left(x)=\frac{1}{\max \left\{1,| n| \right\}}\psi \left(x-n),\hspace{1.0em}\psi \left(x)=\frac{x}{\left(1-{x}^{2})}for n∈Zn\in {\mathbb{Z}}. Let then x=n,y=n+12,z=n+1,x=n,\hspace{1em}y=n+\frac{1}{2},\hspace{1em}z=n+1,so dx(y)=dy(z)=23n,dx(z)=∞.{d}_{x}(y)={d}_{y}\left(z)=\frac{2}{3n},\hspace{1.0em}{d}_{x}\left(z)=\infty .This explains the importance of the dependence of ρd{\rho }_{d}on yy.In some cases, it may happen that we do not know an easy formula for the distance that metrizes the manifold XX.The metrization theorems, such as Nagata-Smirnov metrization theorem and Urysohn’s metrization theorem, are usually proven by showing that there is an embedding of XXinto RN{{\mathbb{R}}}^{{\mathbb{N}}}; to define this embedding, Urysohn’s lemma is exploited to define countably many functions fn:X→[0,1]{f}_{n}:X\to \left[0,1]; then a distance is defined on RN{{\mathbb{R}}}^{{\mathbb{N}}}and pulled back on XX. While perfectly valid as a proof, it is not an easily manageable definition and it is unsuitable for numerical algorithms.If XXis compact, then XXcan be embedded in RN{{\mathbb{R}}}^{N}; so this can be used to define a distance on XX, by carefully tracking how the embedding is defined (as e.g. in §36 in [8]). This plan could be carried on, eventually providing an explicit formula for the distance; in particular, we can assume that the atlas is finite, let #I\#Ibe its cardinality, then such proof provides N=(m+1)#IN=\left(m+1)\#I.We remark that, at the same time, formula (2.14) gives us a very convenient definition of separation functions (also when XXis not compact): those encode the idea of “nearness” and can be used in further proofs and/or for numerical algorithms.3Mild topologyIn this section, we propose a novel topology on the space of continuous functions C0(M;N){C}^{0}\left(M;\hspace{0.33em}N).To define the mild topology we need that (N,dN)\left(N,{d}_{N})be a metric space.We fix a distinguished point p¯∈N\overline{p}\in N.Let f,g∈C0=C0(M;N)f,g\in {C}^{0}={C}^{0}\left(M;\hspace{0.33em}N), we define the “mild separation” (3.1)dfmild,p¯(g)=defsupx∈MdN(f(x),g(x))1+dN(f(x),p¯).{d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\frac{{d}_{N}(f\left(x),g\left(x))}{1+{d}_{N}(f\left(x),\overline{p})}.For f∈C0f\in {C}^{0}and α>0\alpha \gt 0we define the “mild pseudo ball” (3.2)Bmild,p¯(f,α)=def{g∈C0:df(g)<α}.{B}^{\text{mild},\overline{p}}(f,\alpha )\mathop{=}\limits^{{\rm{def}}}\left\{g\in {C}^{0}:{d}_{f}\left(g)\lt \alpha \right\}.We omit the superscripts “mild,p¯\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}” for ease of notation.Definition 3.1The mild topology on C0{C}^{0}is the topology generated by the above sets B(f,α)B(f,\alpha ). We will write CM0(M;N){C}_{M}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.The above definitions will be justified in Proposition 3.5. To this end, we prove these two lemmas as follows.Lemma 3.2(Pseudo symmetry) dg(f)≤θ(df(g)){d}_{g}(f)\le \theta \left({d}_{f}\left(g))with θ(α)=α1−αα<1∞α≥1.\theta \left(\alpha )=\left\{\begin{array}{ll}\frac{\alpha }{1-\alpha }\hspace{1.0em}& \alpha \lt 1\\ \infty \hspace{1.0em}& \alpha \ge 1.\end{array}\right.ProofSuppose 0<α<10\lt \alpha \lt 1and df(g)≤α{d}_{f}\left(g)\le \alpha then dN(f(x),g(x))≤α(1+dN(f(x),p¯))≤α(1+dN(g(x),p¯)+dN(g(x),f(x))),{d}_{N}(f\left(x),g\left(x))\le \alpha (1+{d}_{N}(f\left(x),\overline{p}))\le \alpha (1+{d}_{N}(g\left(x),\overline{p})+{d}_{N}(g\left(x),f\left(x))),hence □(1−α)dN(f(x),g(x))≤α(1+dN(g(x),p¯)).\left(1-\alpha ){d}_{N}(f\left(x),g\left(x))\le \alpha (1+{d}_{N}(g\left(x),\overline{p})).Lemma 3.3(Pseudo triangle inequality) Let f,g,h∈C0f,g,h\in {C}^{0}and α>0\alpha \gt 0; if df(g)≤β<α{d}_{f}\left(g)\le \beta \lt \alpha and dg(h)≤ρd(α,β){d}_{g}\left(h)\le {\rho }_{d}\left(\alpha ,\beta )with(3.3)ρd(α,β)=α−β1+β,{\rho }_{d}\left(\alpha ,\beta )=\frac{\alpha -\beta }{1+\beta },then df(h)≤α{d}_{f}\left(h)\le \alpha .Proofdf(g)≤β{d}_{f}\left(g)\le \beta means dN(f(x),g(x))≤β(1+dN(f(x),p¯)),{d}_{N}(f\left(x),g\left(x))\le \beta (1+{d}_{N}(f\left(x),\overline{p})),moreover, dg(h)≤ρ{d}_{g}\left(h)\le \rho means dN(g(x),h(x))≤ρ(1+dN(g(x),p¯));{d}_{N}\left(g\left(x),h\left(x))\le \rho (1+{d}_{N}\left(g\left(x),\overline{p}));summing them(3.4)dN(f(x),h(x))≤dN(f(x),g(x))+dN(g(x),h(x))≤(β+ρ)+βdN(f(x),p¯)+ρdN(g(x),p¯).{d}_{N}(f\left(x),h\left(x))\le {d}_{N}(f\left(x),g\left(x))+{d}_{N}\left(g\left(x),h\left(x))\le \left(\beta +\rho )+\beta {d}_{N}(f\left(x),\overline{p})+\rho {d}_{N}\left(g\left(x),\overline{p}).At the same time, dN(g(x),p¯)≤dN(g(x),f(x))+dN(f(x),p¯)≤β+(1+β)dN(f(x),p¯):{d}_{N}\left(g\left(x),\overline{p})\le {d}_{N}\left(g\left(x),f\left(x))+{d}_{N}(f\left(x),\overline{p})\le \beta +\left(1+\beta ){d}_{N}(f\left(x),\overline{p}):substituting this in (3.4), dN(f(x),h(x))≤(β+ρ(1+β))+(β+ρ(1+β))dN(f(x),p¯).{d}_{N}(f\left(x),h\left(x))\le (\beta +\rho (1+\beta ))+(\beta +\rho (1+\beta )){d}_{N}(f\left(x),\overline{p}).We just need to find a ρ>0\rho \gt 0such that (β+ρ(1+β))≤α:(\beta +\rho (1+\beta ))\le \alpha :the value defined in equation (3.3) is such a choice.□Remark 3.4Lemma 3.3 proves the pseudo triangle inequality in the form in equation (2.5); then the form in equation (2.2) follows from Proposition 2.10. Moreover, the pseudo reverse triangle inequality (2.3) holds as well, due to Lemma 3.2 and Proposition 2.26. So the family df{d}_{f}defined in formula (3.1) is indeed a family of separation functions, as defined in Definition 2.1.Summarizing, we can state the needed result.Proposition 3.5The pseudo balls B(f,α)B(f,\alpha )are a global base for the mild topology. The mild topology is metrizable.ProofThe first statement follows from Theorem 2.3. The second statement derives from Theorem 2.30; we need to verify the two Hypothesis 2.24. For the first hypothesis, we note that Lemma 3.2 proves that the family of separation functions is pseudo symmetric.The second hypothesis requires that the modulus ρ(y,α,β)\rho (y,\alpha ,\beta )appearing in Definition 2.1 does not depend on yy; this is satisfied by the modulus defined in equation (3.3).□Lemma 3.6The mild topology does not depend on the choice of p¯∈N\overline{p}\in N.ProofGiven p¯,p˜∈N\overline{p},\tilde{p}\in N, for any β>0\beta \gt 0, choosing α=β11+dN(p¯,p˜),\alpha =\beta \frac{1}{1+{d}_{N}\left(\overline{p},\tilde{p})},we have α(1+dN(y,p¯))≤α(1+dN(y,p˜)+dN(p˜,p¯))≤β(1+dN(y,p˜));\alpha \left(1+{d}_{N}(y,\overline{p}))\le \alpha \left(1+{d}_{N}(y,\tilde{p})+{d}_{N}\left(\tilde{p},\overline{p}))\le \beta \left(1+{d}_{N}(y,\tilde{p}));then we reason as in Remark 3.8.□Proposition 3.7The mild topology is stronger than the weak topology; and it is weaker than the strong topology.ProofWe show that the mild topology is stronger than the weak topology. Fix ε>0\varepsilon \gt 0and a compact set K⊆MK\subseteq M, let β=maxx∈KdN(f(x),p¯)\beta =\mathop{\max }\limits_{x\in K}{d}_{N}(f\left(x),\overline{p})and ρ<ε1+β,\rho \lt \frac{\varepsilon }{1+\beta },we know that if g∈B(f,ρ),g\in B(f,\rho ),then ∀x∈K,dN(f(x),g(x))<ε.\forall \hspace{-0.3em}x\in K,{d}_{N}(f\left(x),g\left(x))\lt \varepsilon .The fact that the mild topology is weaker than the strong topology follows from Remark 3.8.□Remark 3.8We may also define the “mild neighborhood” (3.5)B˜(f,α)=def{g∈C0:∀x∈M,dN(f(x),g(x))<α(1+dN(f(x),p¯))}.\tilde{B}(f,\alpha )\mathop{=}\limits^{{\rm{def}}}\left\{g\in {C}^{0}:\forall x\in M,{d}_{N}(f\left(x),g\left(x))\lt \alpha (1+{d}_{N}(f\left(x),\overline{p}))\right\}.Note that, for 0<β<α0\lt \beta \lt \alpha (3.6)B˜(f,β)⊆B(f,α)⊆B˜(f,α),\tilde{B}(f,\beta )\subseteq B(f,\alpha )\subseteq \tilde{B}(f,\alpha ),so “mild neighborhoods” can be used to define the mild topology; unfortunately, they may fail to be open.A “mild neighborhood” can be built using the same method seen in the graph topology (see Definition 1.2): indeed consider open sets of the form U={(x,y)∈M×N:∀x∈M,dN(f(x),y)<α(1+dN(f(x),p¯))}U=\left\{\left(x,y)\in M\times N:\forall x\in M,{d}_{N}(f\left(x),y)\lt \alpha (1+{d}_{N}(f\left(x),\overline{p}))\right\}for f∈C0,α>0f\in {C}^{0},\alpha \gt 0, and then B˜(f,α)={g∈C0:graph(g)∈U}.\tilde{B}(f,\alpha )=\left\{g\in {C}^{0}:\hspace{0.1em}\text{graph}\hspace{0.1em}\left(g)\in U\right\}.Consequently, equation (3.6) proves that the mild topology is coarser than the strong topology.Remark 3.9In general, this topology is not separable. For example, when N=M=RN=M={\mathbb{R}}, setting fs(x)=esx{f}_{s}\left(x)={e}^{sx}, we have dfs(ft)=1s>t∞s<t;{d}_{{f}_{s}}({f}_{t})=\left\{\begin{array}{ll}1\hspace{1.0em}& s\gt t\\ \infty \hspace{1.0em}& s\lt t;\end{array}\right.in these cases the topology does not satisfy the second countability axiom. (This is why we proved Theorem 2.30, that is based on Nagata-Smirnov metrization theorem; we cannot use Urysohn’s metrization theorem to prove that the mild topology is metrizable.)3.1MetrizabilityWe know by Lemma 3.2 and Theorem 2.30 that the mild topology is metrizable.At first sight, a reasonable candidate for a distance that generates the mild topology may be dmild?(f,g)=defsupx∈MdN(f(x),g(x))1+dN(f(x),p¯)+dN(g(x),p¯),{d}_{\text{mild?}}(f,g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\frac{{d}_{N}(f\left(x),g\left(x))}{1+{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),\overline{p})},where f,g∈C0(M;N)f,g\in {C}^{0}\left(M;\hspace{0.33em}N). Note that 0≤dmild?(f,g)<10\le {d}_{\text{mild?}}(f,g)\lt 1.Lemma 3.10Obviously dmild?(f,g)≤dfmild,p¯(g);{d}_{\text{mild?}}(f,g)\le {d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g);indeed the formula defining the LHS has one more positive term in the denominator than the formula defining the RHS. Moreover, for 0<α<10\lt \alpha \lt 1α=dmild?(f,g)⇒dfmild,p¯(g)≤2α1−α.\alpha ={d}_{\text{mild?}}(f,g)\Rightarrow {d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g)\le \frac{2\alpha }{1-\alpha }.ProofIf 0<α<10\lt \alpha \lt 1and dmild?(f,g)≤α{d}_{\text{mild?}}(f,g)\le \alpha , then dN((fx),g(x))≤α(1+dN(f(x),p¯)+dN(g(x),p¯))≤α(1+2dN(f(x),p¯)+dN(g(x),f(x)));{d}_{N}\left((fx),g\left(x))\le \alpha (1+{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),\overline{p}))\le \alpha (1+2{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),f\left(x)));hence □(1−α)dN(f(x),g(x))≤2α(1+dN(f(x),p¯)).\left(1-\alpha ){d}_{N}(f\left(x),g\left(x))\le 2\alpha (1+{d}_{N}(f\left(x),\overline{p})).So if dmild?(f,g){d}_{\text{mild?}}(f,g)is a distance, it will generate the mild topology.But is it a distance? The formula is obviously symmetric, and we have dmild?(f,g)=0⇔f≡g;{d}_{\text{mild?}}(f,g)=0\hspace{0.33em}\iff \hspace{0.33em}f\equiv g;the question is as follows: does it satisfy the triangle inequality?Consider then this formula dN?(z,w)=defdN(z,w)1+dN(z,p¯)+dN(w,p¯){d}_{\text{N?}}\left(z,w)\mathop{=}\limits^{{\rm{def}}}\frac{{d}_{N}\left(z,w)}{1+{d}_{N}(z,\overline{p})+{d}_{N}(w,\overline{p})}for z,w∈Nz,w\in N; so dmild?(f,g)=supx∈MdN?(f(x),g(x)).{d}_{\text{mild?}}(f,g)=\mathop{\sup }\limits_{x\in M}{d}_{\text{N?}}(f\left(x),g\left(x)).We note that dmild?(f,g){d}_{\text{mild?}}(f,g)satisfies the triangle inequality if and only if dN?(z,w){d}_{\text{N?}}\left(z,w)does. (For one implication, consider constant functions; for the other, use standard properties of the supremum.)Unfortunately, the quantity dN?(z,w){d}_{\text{N?}}\left(z,w)does not satisfy the triangle inequality for some choices of NN; as is seen in this example: let NNbe a circle of length 13 where the points are posed as in Figure 3.Figure 3Points along a circle NNof length 13 and distances.Remark 3.11At the same time, consider the case when NNis a Hilbert space and p¯=0\overline{p}=0, then dN?(z,w)=def‖z−w‖N1+‖z‖N+‖w‖N.{d}_{\text{N?}}\left(z,w)\mathop{=}\limits^{{\rm{def}}}\frac{\Vert z-w{\Vert }_{N}}{1+\Vert z{\Vert }_{N}+\Vert w{\Vert }_{N}}.Note that the formula is invariant for rotations, so it is enough to check the triangle inequality for N=R3N={{\mathbb{R}}}^{3}; numerical experiments suggest that it is indeed a distance; to this end, we tested the triangle inequality with randomly sampled points and tried to numerically minimize the difference dN?(x,y)+dN?(y,z)−dN?(x,z){d}_{\text{N?}}\left(x,y)+{d}_{\text{N?}}(y,z)-{d}_{\text{N?}}\left(x,z)for x,y,z∈R3x,y,z\in {{\mathbb{R}}}^{3}. We though could not prove it analytically. See addendum material for more information.3.2Properties of proper mapsLemma 3.12Suppose that (N,dN)\left(N,{d}_{N})is a “proper metric space,” i.e., closed balls are compact.If 0<α<10\lt \alpha \lt 1and f∈B(g,α)f\in B\left(g,\alpha ), then ggis proper iff f is proper.Similarly for the “mild neighborhood” B˜(f,α)\tilde{B}(f,\alpha )defined in Remark 3.8.ProofSuppose that dg(f)=D<∞{d}_{g}(f)=D\lt \infty and ffis proper, we prove that ggis proper. Let K⊆NK\subseteq Nbe compact, let R=maxy∈KdN(y,p¯),R=\mathop{\max }\limits_{y\in K}{d}_{N}(y,\overline{p}),and let H={y∈N:dN(y,p¯)≤D+R(D+1)};H=\{y\in N:{d}_{N}(y,\overline{p})\le D+R\left(D+1)\};then HHis compact. We prove that g−1(K)⊆f−1(H),{g}^{-1}\left(K)\subseteq {f}^{-1}\left(H),so that g−1(K){g}^{-1}\left(K)is compact. Indeed, if x∈g−1(K)x\in {g}^{-1}\left(K), then g(x)∈Kg\left(x)\in Kso dN(g(x),p¯)≤R{d}_{N}\left(g\left(x),\overline{p})\le R, hence dN(f(x),p¯)≤dN(g(x),p¯)+dN(g(x),f(x))≤D+(D+1)dN(g(x),p¯)≤D+R(D+1),{d}_{N}(f\left(x),\overline{p})\le {d}_{N}\left(g\left(x),\overline{p})+{d}_{N}\left(g\left(x),f\left(x))\le D+\left(D+1){d}_{N}\left(g\left(x),\overline{p})\le D+R\left(D+1),so f(x)∈Hf\left(x)\in H. Suppose now that ggis proper and dg(f)<1{d}_{g}(f)\lt 1, then, by pseudo symmetry Lemma 3.2, df(g)<∞{d}_{f}\left(g)\lt \infty . So ffis proper.□Corollary 3.13The set of proper maps is both open and closed in the mild topology.3.3Properties of affine actionsTheorem 3.14Suppose that N=RnN={{\mathbb{R}}}^{n}, dN(x,y)=∣y−x∣{d}_{N}\left(x,y)=| y-x| is the usual Euclidean distance; endow C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})with the mild topology; then the actions listed in Section 1.5 are (jointly) continuous.(The proof is in page 23 in Section A.4.)Remark 3.15For some specific actions some extra information may be useful. Rotation. If we choose p¯=0\overline{p}=0for convenience in the definition equation (3.2) (as is made possible by Lemma 3.6), then, given a rotation R∈O(n)R\in O\left(n), the map f∈C0↦Rf∈C0f\in {C}^{0}\mapsto Rf\in {C}^{0}is an “isometry”: indeed, B(Rf,α)=RB(f,α)B\left(Rf,\alpha )=RB(f,\alpha )because dRfRg=dfg.{d}_{Rf}Rg={d}_{f}g.We also note that for S,R∈O(n)S,R\in O\left(n), ∣Rg(x)−Sg(x)∣≤‖R−S‖∣g(x)∣;| Rg\left(x)-Sg\left(x)| \le \Vert R-S\Vert | g\left(x)| ;so dRg(Sg)≤‖R−S‖{d}_{Rg}\left(Sg)\le \Vert R-S\Vert where ‖R−S‖\Vert R-S\Vert is a matrix (operator) norm.Rescaling. Let s>0s\gt 0, let m=min{1,s},M=max{1,s}m=\min \left\{1,s\right\},M=\max \left\{1,s\right\}then mdf(g)≤dsf(sg)≤Mdf(g),m{d}_{f}\left(g)\le {d}_{sf}\left(sg)\le M{d}_{f}\left(g),so f∈C0↦sf∈C0f\in {C}^{0}\mapsto sf\in {C}^{0}is again a homeomorphism. For the action s∈R↦sf∈C0s\in {\mathbb{R}}\mapsto sf\in {C}^{0}similarly ∣t−s∣m≤dsf(tf)≤∣t−s∣M.| t-s| m\le {d}_{sf}\left(tf)\le | t-s| M.3.4CaveatsWe have then seen many good properties of the mild topology; there are some drawbacks though. The mild topology depends on the choice of distance dN{d}_{N}.It is not invariant w.r.t. homeomorphisms as in Proposition 1.4; it is invariant only for right action, i.e., if ΦM:M˜→M{\Phi }_{M}:\tilde{M}\to Mis a homeomorphism, then the map f↦f∘ΦMf\mapsto f\circ {\Phi }_{M}is a homeomorphism between C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)and C0(M˜;N){C}^{0}\left(\tilde{M};\hspace{0.33em}N), where both spaces are endowed with the mild topology.The space C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)with the mild topology may fail to be connected, since proper maps are open and closed.When N=RnN={{\mathbb{R}}}^{n}with Euclidean structure, the space C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})with the mild topology is not in general a topological vector space; there are many reasons, we list some. For g∈C0g\in {C}^{0}fixed, the map f∈C0↦g+f∈C0f\in {C}^{0}\mapsto g+f\in {C}^{0}may fail to be continuous. For example, consider g(x)=−exg\left(x)=-{e}^{x}and C0=C0(R;R){C}^{0}={C}^{0}\left({\mathbb{R}};{\mathbb{R}}), f(x)=exf\left(x)={e}^{x}. Then the counter image of the mild pseudo ball B(0,1)B\left(0,1)is {h(x)+ex:h∈C0,supx∈Rn∣h(x)∣<1},\left\{\phantom{\rule[-1.25em]{}{0ex}}h\left(x)+{e}^{x}:h\in {C}^{0},\mathop{\sup }\limits_{x\in {{\mathbb{R}}}^{n}}| h\left(x)| \lt 1\right\},and it does not contain any mild pseudo ball B(ex,ε)B\left({e}^{x},\varepsilon ).For f∈C0f\in {C}^{0}fixed, the map λ∈R↦λf∈C0\lambda \in {\mathbb{R}}\mapsto \lambda f\in {C}^{0}may fail to be continuous at λ=0\lambda =0(adapting the previous example).The space may not be connected.4ConclusionWe have discussed a novel method to define topologies, by separation functions; we have shown that, even when the topology happens to be metrizable, it may happen that the actual metric is not known and/or that the separation functions are more manageable than the metric that metrizes the topology.We have studied the mild topology C0(M;N){C}^{0}\left(M;\hspace{0.33em}N); it has some good properties: proper maps are a closed and open subset of C0(M;N){C}^{0}\left(M;\hspace{0.33em}N), as in the case of the strong topology; affine actions on N=RnN={{\mathbb{R}}}^{n}are continuous on C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}), as in the case of the weak topology.It is possible to define similar concepts for Cr(M;N){C}^{r}\left(M;\hspace{0.33em}N), the space of rrtimes differentiable maps between two differentiable manifolds M,NM,N; similar properties hold and can be extended to other interesting classes of maps such as immersions, free immersions, submersions, embeddings, diffeomorphisms; this may be argument of a forthcoming paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

Separation functions and mild topologies

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de Gruyter
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© 2023 the author(s), published by De Gruyter
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2299-3274
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10.1515/agms-2022-0149
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Abstract

1IntroductionLet MMand NNbe Hausdorff topological spaces, with topologies τM{\tau }_{M}and τN{\tau }_{N}. (Sometimes we may also assume that MMor NNbe metric spaces with distances dM{d}_{M}, or respectively dN{d}_{N}; then τM{\tau }_{M}, or respectively τN{\tau }_{N}, will be the associated topology.)In Section 3, we discuss a topology for the space C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)of continuous maps f:M→Nf:M\to N, which we will call “mild topology.”To define and discuss the properties of the mild topology, we have developed a novel method whereby we define a topology τ\tau on a generic set XXby using a family of “separation functions;” this method is presented in Section 2. Separation functions are somewhat similar to the usual distance function in metric spaces (M,d)\left(M,d), but they have weaker hypotheses (so they can be more manageable in some contexts). Separation functions are used to define pseudo balls that are a global base for a T2 topology τ\tau (this is proven in Theorem 2.3). Under some additional hypotheses, we will define “set separation functions” (similar to set distance functions) in Section 2.5 to prove that the topology is T6. Moreover, under further hypotheses, the topology τ\tau is, in fact, metrizable (Theorems 2.21 and 2.30) – such will be the case for the mild topology. We will also discuss other applications of this separation function method. In Section 2.3, we show that the topology on the Sorgenfrey line (that is not metrizable) can be defined by using a suitable family of separation functions. In Section 2.12, we show how separation functions can be easily defined for topological manifolds starting from the atlas of the manifold. These examples show that the method of separation functions is a promising tool that may have further uses beyond the definition of mild topology.After developing this method of separation functions in Section 2, we then proceed, in Section 3, to apply it to the case X=C0(M;N)X={C}^{0}\left(M;\hspace{0.33em}N)to define the mild topology.But first and foremost, we would like to explain why we may find useful a new topology on C0(M;N){C}^{0}\left(M;\hspace{0.33em}N). For this purpose, in Section 1.1 we recapitulate the definitions of “strong” and “weak” topologies; in Section 1.2, assuming N=RnN={{\mathbb{R}}}^{n}, we relate those to the usual definition of Frechét spaces C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}); then, in Sections 1.3, 1.4 and 1.5, we express some properties of these topologies.1.1Topologies for continuous mapsWe define the “graph” of the function f:M→Nf:M\to Nas graph(f)=def{(x,y)∈M×N:y=f(x)}.\hspace{0.1em}\text{graph}\hspace{0.1em}(f)\mathop{=}\limits^{{\rm{def}}}\left\{\left(x,y)\in M\times N:y=f\left(x)\right\}.We distinguish two fundamental examples of topologies for C0=C0(M;N){C}^{0}={C}^{0}\left(M;\hspace{0.33em}N).Definition 1.1The compact-open topology is generated by sets of the form {f∈C0:f(K)⊆U},\{f\in {C}^{0}:f\left(K)\subseteq U\},where K⊆MK\subseteq Mis compact and U⊆NU\subseteq Nis open. (This collection of sets is a subbase for the topology, but it does not always form a base for a topology.) It is also called the “topology of uniform convergence on compact sets” or the “weak topology” in [2]. We will write CW0(M;N){C}_{W}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.Definition 1.2The graph topology is generated by sets of the form {f∈C0(M;N):graph(f)⊆U},\{f\in {C}^{0}\left(M;\hspace{0.33em}N):\hspace{0.1em}\text{graph}\hspace{0.1em}(f)\subseteq U\},where UUruns through all open sets in M×NM\times N. It is also called “wholly open topology” in [4], “fine” or “Whitney” or “strong topology” in [2]. We will write CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.An equivalent definition of the strong topology can be formulated under additional hypotheses.Proposition 1.3(41.6 in [4], or Chapter 2 Section 4 in [2]) If M is paracompact and (N,dN)\left(N,{d}_{N})is a metric space, then for f∈C0(M,N)f\in {C}^{0}\left(M,N)the sets {g∈C0(M,N):dN(g(x),f(x))<ε(x)∀x∈M}\left\{g\in {C}^{0}\left(M,N):{d}_{N}\left(g\left(x),f\left(x))\lt \varepsilon \left(x)\forall x\in M\right\}form a base of neighborhoods for the graph topology, where ε\varepsilon runs through all positive continuous functions on M.Another way to state this resultExercise 3 in Chapter 2, Section 4 in [2], to be compared with the “counterexample 1.1.8” in [1].is to define the distances dε(f,g)=defsupx∈Mε(x)dN(g(x),f(x));{d}_{\varepsilon }(f,g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\varepsilon \left(x){d}_{N}\left(g\left(x),f\left(x));then the topology generated by all these distances is the graph topology.It is possible to define similar concepts for Cr(M;N){C}^{r}\left(M;\hspace{0.33em}N), the space of rrtimes differentiable maps between two differentiable manifolds M,NM,N. In this case, we do not detail the discussion.The aforementioned topologies are invariant in the sense that the next proposition shows.Proposition 1.4If ΦM:M˜→M{\Phi }_{M}:\tilde{M}\to Mand ΦN:N→N˜{\Phi }_{N}:N\to \tilde{N}are homeomorphisms, then the map f↦ΦN∘f∘ΦMf\mapsto {\Phi }_{N}\circ f\circ {\Phi }_{M}is a homeomorphism between C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)and C0(M˜;N˜){C}^{0}\left(\tilde{M};\hspace{0.33em}\tilde{N}), where the spaces are both endowed either with the “weak” or the “strong” topology.1.2When N=RnN={{\mathbb{R}}}^{n}Let us suppose in this section that NNis the standard Euclidean space Rn{{\mathbb{R}}}^{n}and that MMis a Hausdorff locally compact and second countableThat is, it admits a countable base of open sets.topological space.We recall that any second countable Hausdorff space that is locally compact is paracompact, so Proposition 1.3 applies in the current context; moreover, there exists a countable locally finite covering of open sets, each with compact closure.Definition 1.5Cloc0(M,Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M,{{\mathbb{R}}}^{n})is the Frechét space whose topology is generated by the seminorms [f]K=supx∈K∣f(x)∣,{[f]}_{K}=\mathop{\sup }\limits_{x\in K}| f\left(x)| ,for K⊆MK\subseteq Mcompact. If MMis compact, then it coincides with the usual Banach space C0(M,Rn){C}^{0}\left(M,{{\mathbb{R}}}^{n})associated with the norm ‖f‖=supx∈M∣f(x)∣.\Vert f\Vert =\mathop{\sup }\limits_{x\in M}| f\left(x)| .Proposition 1.6Under the above hypotheses, CW0(M;Rn){C}_{W}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})coincides with Cloc0(M,Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M,{{\mathbb{R}}}^{n}).We can define also another topology.For K⊆MK\subseteq M, compact we define the subset V0,K={f∈C0(M;Rn):supp(f)⊆K}{V}_{0,K}=\{f\in {C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}):\hspace{0.1em}\text{supp}\hspace{0.1em}(f)\subseteq K\}of C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})functions with support in KK. Each such V0,K{V}_{0,K}is a closed subspace of Cloc0(M;Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}). Then Cloc0(M;Rn){C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})is a Frechét space with the induced topology.We can then define this topology.Definition 1.7(Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology) The Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology is the strict inductive limitFor the definition of strict inductive limit and its properties, we refer to 17G at page 148 in [3].with respect to the inclusions V0,K→C0(M;Rn){V}_{0,K}\to {C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})for K⊆MK\subseteq Mcompact. A set WWis open in the Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})topology if, for all K⊆MK\subseteq Mcompact, W∩V0,KW\cap {V}_{0,K}is open in V0,K{V}_{0,K}.1.3Properties of weak and strong topologiesThe weak topology enjoys nice metrization properties.Proposition 1.8Let N be metrizable with a complete metric, and let MMbe locally compact and second countable. Then CW0(M,N){C}_{W}^{0}\left(M,N)has a complete metric.This is proven in Theorem 4.1 in Chapter 2, Section 4 in [2].We ponder on these remarks, taken from [2].Remark 1.9The topological space CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)resulting from the strong topology is the same as CW0(M;N){C}_{W}^{0}\left(M;\hspace{0.33em}N)if MMis compact. If MMis not compact, however, CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)can be an extremely large topology; for example, when M,NM,Nare differentiable finite dimensional manifolds (of positive dimension), then it is not metrizable and in fact it does not have a countable local base at any point, and it has uncountably many connected components.In particular, if MMis not compact, CS0(M;Rn){C}_{S}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})cannot be a topological vector space, but the connected component containing f≡0f\equiv 0coincides with Cc0(M;Rn){C}_{c}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}).1.4Proper mapsDefinition 1.10A proper map f:M→Nf:M\to Nis a continuous map such that we have that f−1(K){f}^{-1}\left(K)is compact in MM, for any K⊆NK\subseteq Ncompact.Obviously, if MMis compact then any continuous function is proper. More in general:Lemma 1.11The set of proper maps f:M→Nf:M\to Nis both open and closed in the strong CS0(M;N){C}_{S}^{0}\left(M;\hspace{0.33em}N)topology.For the proof see, e.g., Section 5.1 in [7] or Theorem 1.5 in Chapter 2, Section 1 in [2].It is easily seen that, in general, the set of proper functions is neither closed nor open in any Clocr{C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{r}topology.Example 1.12Consider functions f:R→Rf:{\mathbb{R}}\to {\mathbb{R}}, and the weak topology; it is easy to show examples of sequences fn→f{f}_{n}\to fsuch that none of the fn{f}_{n}are proper, but ffis, e.g., fn(x)=narctan(x/n){f}_{n}\left(x)=n\arctan \left(x\hspace{0.1em}\text{/}\hspace{0.1em}n)and f(x)=xf\left(x)=x;fn{f}_{n}are all proper but ffis not, e.g., fn(x)=1nx2{f}_{n}\left(x)=\frac{1}{n}{x}^{2}and f(x)=0f\left(x)=0.The aforementioned examples hold also in Cloc∞{C}_{\hspace{0.1em}\text{loc}\hspace{0.1em}}^{\infty }, where convergence is defined as “local uniform convergence of all derivatives.”1.5Drawback of strong topologiesGiven the above discussion, it would seem that, when dealing with proper maps, it would be better to use a strong topology. Strong topologies have drawbacks as well.In particular, consider the case of maps f:M→Rnf:M\to {{\mathbb{R}}}^{n}, let C0=C0(M;Rn){C}^{0}={C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}); we have some natural actions: translations, given by the action (v,f)∈Rn×C0↦f+v∈C0;\left(v,f)\in {{\mathbb{R}}}^{n}\times {C}^{0}\mapsto f+v\in {C}^{0};rotations (A,f)∈SO(n)×C0↦Af∈C0;\left(A,f)\in {\rm{\text{SO}}}\left(n)\times {C}^{0}\mapsto Af\in {C}^{0};rescalings (λ,f)∈I×C0↦λf∈C0,\left(\lambda ,f)\in I\times {C}^{0}\mapsto \lambda f\in {C}^{0},for I=(0,∞)⊂RI=\left(0,\infty )\subset {\mathbb{R}}and, in general, affine transformations F,f↦Ff,F,f\mapsto Ff,where Fy=Ay+vFy=Ay+vis given by A∈GL(Rn)A\in \hspace{0.1em}\text{GL}\hspace{0.1em}\left({{\mathbb{R}}}^{n}), v∈Rnv\in {{\mathbb{R}}}^{n}.These actions are continuous if C0{C}^{0}is endowed with the weak topology, but it may fail to be continuous if C0{C}^{0}is endowed with the strong topology.There is another drawback (that we already remarked): when N=RnN={{\mathbb{R}}}^{n}, then C0{C}^{0}with the strong topology may fail to be a topological vector space.We will show in Section 3 that the mild topology shares some good properties valid for the weak and the strong topology.2Topology by separation functionsIn this section, given a generic set XX, we will use “separation functions” to define a topology τ\tau on XX; we will then study the properties of this topology.Definition 2.1A family d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}of real positive functions dx(y):X→[0,∞],{d}_{x}(y):X\to \left[0,\infty ],one for each x∈Xx\in X, is a family of separation functions when dx(y)=0{d}_{x}(y)=0iff x=yx=y;given y∈Xy\in Xand α,β∈R\alpha ,\beta \in {\mathbb{R}}with 0<β<α0\lt \beta \lt \alpha there exists a function (2.1)ρd=ρd(y,α,β)>0{\rho }_{d}={\rho }_{d}(y,\alpha ,\beta )\gt 0(called “modulus”) such that, for all x,z∈Xx,z\in X, (2.2)dx(y)≤β∧dy(z)<ρd⇒dx(z)<α,{d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,(2.3)dx(y)≥α∧dy(z)<ρd⇒dx(z)>β.{d}_{x}(y)\ge \alpha \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\gt \beta .This condition (2.2) will be called “pseudo triangle inequality,” while condition (2.3) will be called “pseudo reverse triangle inequality” (Figure 1).Figure 1Representation of triangle inequalities.These are written as dx(y){d}_{x}(y)and not d(x,y)d\left(x,y)to remark that they do not satisfy the axioms of “distances”: they are not required to be symmetric and do not satisfy the standard triangle inequality.Remark 2.2An “asymmetric distance” (a.k.a. “quasi metric”) is a function b(x,y):X2→[0,+∞]b\left(x,y):{X}^{2}\to \left[0,+\infty ]that satisfies the separation property “b(x,y)=0⇔x=yb\left(x,y)=0\hspace{0.33em}\iff \hspace{0.33em}x=y” and the standard triangle inequality, but it may fail to be symmetric. See [5,6] and references therein. An “asymmetric distance” immediately provides a family of separation functions dx(y)=b(x,y){d}_{x}(y)=b\left(x,y)with ρd(y,α,β)=α−β{\rho }_{d}(y,\alpha ,\beta )=\alpha -\beta .Theorem 2.3Given a set X with a family of separation functions, then we can define “pseudo balls” B(x,ε)={y∈X:dx(y)<ε};B\left(x,\varepsilon )=\{y\in X:{d}_{x}(y)\lt \varepsilon \};these are then a global base for a T2 topology τ\tau , and each B(x,ε)B\left(x,\varepsilon )is an open neighborhood of xxin (X,τ)\left(X,\tau ).ProofIndeed the pseudo triangle inequality (2.2) shows that if y∈B(x,α)y\in B\left(x,\alpha )and β=dx(y)\beta ={d}_{x}(y)and ρd=ρd(y,α,β){\rho }_{d}={\rho }_{d}(y,\alpha ,\beta ), then B(y,ρd)⊆B(x,α).B(y,{\rho }_{d})\subseteq B\left(x,\alpha ).The pseudo reverse triangle inequality (2.3) shows that if dx(y)=α>0{d}_{x}(y)=\alpha \gt 0then □B(x,β)∩B(y,ρd)=∅.B\left(x,\beta )\cap B(y,{\rho }_{d})=\varnothing .Proposition 2.4Each dx(y){d}_{x}(y), for fixed x, is continuous on (X,τ)\left(X,\tau ).ProofTheorem 2.3 readily implies that dx(y){d}_{x}(y)is upper semicontinuous: indeed we already know that B(x,ε)={y∈X:dx(y)<ε}B\left(x,\varepsilon )=\{y\in X:{d}_{x}(y)\lt \varepsilon \}is open. Let V={z∈X:dx(z)>ε};V=\left\{z\in X:{d}_{x}\left(z)\gt \varepsilon \right\};we want to prove that it is open. Let y∈Vy\in V, let α=dx(y)>ε\alpha ={d}_{x}(y)\gt \varepsilon , and let ε<β<α\varepsilon \lt \beta \lt \alpha : the pseudo reverse triangle inequality (2.3) shows that □B(y,ρd)⊆V.B(y,{\rho }_{d})\subseteq V.It is interesting to note that separation functions have a form of stability that makes them more manageable than distance functions.Proposition 2.5Suppose that dx(y){d}_{x}(y)is a separation function and φ:[0,∞]→[0,∞]\varphi :\left[0,\infty ]\to \left[0,\infty ]is in the Θ\Theta class (defined in Definition 2.11); let bx(y)=φ(dx(y)),{b}_{x}(y)=\varphi \left({d}_{x}(y)),then bx(y){b}_{x}(y)is a separation function.Remark 2.6A similar proposition holds for distances when φ\varphi is also subadditive. But dx(y)=∣x−y∣2{d}_{x}(y)=| x-y{| }^{2}is a separation function on R{\mathbb{R}}, and it is not a distance.Remark 2.7Note that we did not assume validity of this statement.(2.4)“Given x∈X and α>0 for any 0<β<α there exists ρ>0 such thatdx(y)<ρ∧dy(z)≤β⇒dx(z)<α.”\begin{array}{l}\hspace{0.1em}\text{&#x201C;Given\hspace{0.5em}}x\in X\text{\hspace{0.5em}and\hspace{0.5em}}\alpha \gt 0\text{\hspace{0.5em}for any\hspace{0.5em}}0\lt \beta \lt \alpha \text{\hspace{0.5em}there\hspace{0.5em}exists\hspace{0.5em}}\rho \gt 0\text{\hspace{0.5em}such that}\hspace{0.1em}\\ {d}_{x}(y)\lt \rho \wedge {d}_{y}\left(z)\le \beta \Rightarrow {d}_{x}\left(z)\lt \alpha \hspace{0.1em}\text{.&#x201D;}\end{array}\hspace{8em}(and similarly for a “reverse” version).This raises an (yet) unanswered question. Let us define d˜y(x)=dx(y):{\tilde{d}}_{y}\left(x)={d}_{x}(y):under which conditions d˜y(x){\tilde{d}}_{y}\left(x)is a separation function?2.1On the modulusWe recall that the function ρd{\rho }_{d}defined in equation (2.1) is called “modulus.” We now prove that, given a family of separation functions, there exists a maximum modulus.Proposition 2.8Having fixed a family of separation functions, there exists a maximum modulus ρˆd{\hat{\rho }}_{d}, which can be explicitly defined as follows (for y∈Xy\in Xand 0<β<α0\lt \beta \lt \alpha ): ρˆd(y,α,β)=defmax{r∈[0,∞]:∀x,z∈X,(dx(y)≤β∧dy(z)<r⇒dx(z)<α)∧(dx(y)≥α∧dy(z)<r⇒dx(z)>β)}.{\hat{\rho }}_{d}(y,\alpha ,\beta )\mathop{=}\limits^{{\rm{def}}}\max \{r\in \left[0,\infty ]:\forall x,z\in X,\left({d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\lt r\Rightarrow {d}_{x}\left(z)\lt \alpha )\wedge ({d}_{x}(y)\ge \alpha \wedge {d}_{y}\left(z)\lt r\Rightarrow {d}_{x}\left(z)\gt \beta )\}.ProofThe fact that ρˆd{\hat{\rho }}_{d}is a “modulus” is obvious from the definition, since the set on right-hand side (RHS) encodes the “pseudo triangle inequality” (2.2) and the “pseudo reverse triangle inequality” (2.3).We prove that the formula defining ρˆd{\hat{\rho }}_{d}is correct, i.e., that the set on RHS has positive maximum; to this end, we rewrite it in this form ρˆd(y,α,β)=defmaxA(y,α,β),A(y,α,β)=⋂x,z∈XP(y,α,β,x,z)∩R(y,α,β,x,z),P(y,α,β,x,z)=def{r∈[0,∞]:(dx(y)>β∨dy(z)≥r∨dx(z)<α)},R(y,α,β,x,z)=def{r∈[0,∞]:(dx(y)<α∨dy(z)≥r∨dx(z)>β)};\begin{array}{rcl}{\hat{\rho }}_{d}(y,\alpha ,\beta )& \mathop{=}\limits^{{\rm{def}}}& \max {A}_{(y,\alpha ,\beta )},\\ {A}_{(y,\alpha ,\beta )}& =& \bigcap _{x,z\in X}{P}_{(y,\alpha ,\beta ,x,z)}\cap {R}_{(y,\alpha ,\beta ,x,z)},\\ {P}_{(y,\alpha ,\beta ,x,z)}& \mathop{=}\limits^{{\rm{def}}}& \{r\in \left[0,\infty ]:\left({d}_{x}(y)\gt \beta \vee {d}_{y}\left(z)\ge r\vee {d}_{x}\left(z)\lt \alpha )\},\\ {R}_{(y,\alpha ,\beta ,x,z)}& \mathop{=}\limits^{{\rm{def}}}& \{r\in \left[0,\infty ]:({d}_{x}(y)\lt \alpha \vee {d}_{y}\left(z)\ge r\vee {d}_{x}\left(z)\gt \beta )\};\end{array}then we express the last two terms as P(y,α,β,x,z)=⋂dx(y)≤β∧dx(z)≥α[0,dy(z)],R(y,α,β,x,z)=⋂dx(y)≥α∧dx(z)≤β[0,dy(z)];\begin{array}{rcl}{P}_{(y,\alpha ,\beta ,x,z)}& =& \bigcap _{{d}_{x}(y)\le \beta \wedge {d}_{x}\left(z)\ge \alpha }\left[0,{d}_{y}\left(z)],\\ {R}_{(y,\alpha ,\beta ,x,z)}& =& \bigcap _{{d}_{x}(y)\ge \alpha \wedge {d}_{x}\left(z)\le \beta }\left[0,{d}_{y}\left(z)];\end{array}eventually we write A(y,α,β)=[0,∞]∩(⋂x,z∈X,dx(y)≤β∧dx(z)≥α[0,dy(z)])∩(⋂x,z∈X,dx(y)≥α∧dx(z)≤β[0,dy(z)]).{A}_{(y,\alpha ,\beta )}=\left[0,\infty ]\cap \left(\bigcap _{x,z\in X,{d}_{x}(y)\le \beta \wedge {d}_{x}\left(z)\ge \alpha }\left[0,{d}_{y}\left(z)]\right)\cap \left(\bigcap _{x,z\in X,{d}_{x}(y)\ge \alpha \wedge {d}_{x}\left(z)\le \beta }\left[0,{d}_{y}\left(z)]\right).This latter is an intersection of closed intervals starting from zero (included), hence A(y,α,β){A}_{(y,\alpha ,\beta )}is a closed interval of the form [0,ρ]\left[0,\rho ]; moreover, each interval in the RHS contains ρd(y,α,β){\rho }_{d}(y,\alpha ,\beta ), so the maximum is positive.□Remark 2.9It is clear by the aforementioned formulas that ρˆd{\hat{\rho }}_{d}is weakly increasing in α\alpha and weakly decreasing in β\beta ; so we may assume this in Definition 2.1, with no loss of generality.If we add strict monotonicity and continuity, we obtain an interesting proposition.Proposition 2.10Consider the “pseudo triangle inequality” equation (2.2) and these three additional conditions: ∀x,y,z∈X,∀α,β\forall x,y,z\in X,\forall \alpha ,\beta with 0<β<α0\lt \beta \lt \alpha , (2.5)dx(y)≤β∧dy(z)≤ρd⇒dx(z)≤α,{d}_{x}(y)\le \beta \wedge {d}_{y}\left(z)\le {\rho }_{d}\Rightarrow {d}_{x}\left(z)\le \alpha ,(2.6)dx(y)<β∧dy(z)≤ρd⇒dx(z)<α,{d}_{x}(y)\lt \beta \wedge {d}_{y}\left(z)\le {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,(2.7)dx(y)<β∧dy(z)<ρd⇒dx(z)<α,{d}_{x}(y)\lt \beta \wedge {d}_{y}\left(z)\lt {\rho }_{d}\Rightarrow {d}_{x}\left(z)\lt \alpha ,where again ρd=ρd(y,α,β){\rho }_{d}={\rho }_{d}(y,\alpha ,\beta )is the same function (2.1) as used in equation (2.2).In the above four formulas (2.2), (2.5), (2.6), and (2.7), we have alternated strict and loose inequalities.Suppose that the function ρd(y,α,β){\rho }_{d}(y,\alpha ,\beta ), for fixed yy, is continuous in α,β\alpha ,\beta , strictly increasing in α\alpha and strictly decreasing in β\beta : then the four conditions (2.2), (2.5), (2.6), and (2.7) are equivalent.(The proof is in Section A.1 in page 19)This happens for distances, where ρd=α−β{\rho }_{d}=\alpha -\beta ; and it happens in Section 3.A similar statement holds for the “pseudo reverse triangle inequality” (2.3): we skip it for brevity.2.2Fundamental familyIn the aforementioned sections, we can assume that there are many families of separation functions di=(dx,i)x∈X{d}_{i}={\left({d}_{x,i})}_{x\in X}, for i∈Ii\in Ia family of indexes (not depending on xx); this is analogous to the framework in locally convex topological vector spaces, where we have multiple seminorms that are used to define multiple balls centered at zero (and then translated to all other points).But, in the following considerations, we will assume for simplicity that, for each x∈Xx\in X, there is only one separation function. So we have a fundamental family of separation functions d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}. Then the topology satisfies the first countability axiom.2.3The Sorgenfrey lineThe Sorgenfrey line is the set X=RX={\mathbb{R}}with the topology generated by all the half-open intervals [a,b)\left[a,b), where a,b∈R,a<ba,b\in {\mathbb{R}},a\lt b; this family is a global base for the topology. See example 51 in [10].We can define (2.8)b(x,y)=y−xy≥x+∞y<x,b\left(x,y)=\left\{\begin{array}{ll}y-x\hspace{1.0em}& y\ge x\\ +\infty \hspace{1.0em}& y\lt x,\end{array}\right.this is an “asymmetric distance” that generates the above topology on R{\mathbb{R}}; by Remark 2.2 dx(y)=b(x,y){d}_{x}(y)=b\left(x,y)is a fundamental family of separation functions, with ρd(y,α,β)=α−β{\rho }_{d}(y,\alpha ,\beta )=\alpha -\beta .The Sorgenfrey line is a T6 space (a perfectly normal Hausdorff space); it is first countable and separable, but not second countable, so it is not metrizable.This suggests that we need some extra hypotheses to obtain better properties.2.4Pseudo symmetryWe define a convenient class.Definition 2.11(Class Θ\Theta ) We define the class Θ\Theta of functions θ:[0,∞]→[0,∞]\theta :\left[0,\infty ]\to \left[0,\infty ]that are continuous, have θ(0)=0\theta \left(0)=0, and are strictly increasing where they are finite. For such a θ\theta we agree that θ−1∈Θ{\theta }^{-1}\in \Theta is so defined (with a slight abuse of notation) θ−1(s)=defsup{t∈[0,∞):θ(t)<s}.{\theta }^{-1}\left(s)\mathop{=}\limits^{{\rm{def}}}\sup \left\{t\in \left[0,\infty ):\theta \left(t)\lt s\right\}.Equivalently, if D=sup0≤s<∞θ(s),D=\mathop{\sup }\limits_{0\le s\lt \infty }\theta \left(s),then θ−1(s){\theta }^{-1}\left(s)is the usual inverse for s<Ds\lt D, otherwise θ−1(s)=+∞{\theta }^{-1}\left(s)=+\infty Indeed any s≥Ds\ge Dcannot be equal to θ(t)\theta \left(t)for t<∞t\lt \infty , since θ\theta is strictly increasing.. This implies that t=θ−1(s)⇔s=θ(t),t={\theta }^{-1}\left(s)\hspace{0.33em}\iff \hspace{0.33em}s=\theta \left(t),whenever s,t<∞s,t\lt \infty .Remark 2.12This is just a convenient choice, which allows us to simplify notation and analysis. For example, the “tangent” function can be represented in Θ\Theta by defining it as θ(t)=tan(t)s<π/2+∞s≥π/2,\theta \left(t)=\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}\tan \left(t)\hspace{1.0em}& s\lt \pi \hspace{0.1em}\text{/}\hspace{0.1em}2\\ +\infty \hspace{1.0em}& s\ge \pi \hspace{0.1em}\text{/}\hspace{0.1em}2,\end{array}\right.then its “inverse” is just θ−1(s)=arctan(s){\theta }^{-1}\left(s)=\arctan \left(s), with θ−1(+∞)=π/2{\theta }^{-1}\left(+\infty )=\pi \hspace{0.1em}\text{/}\hspace{0.1em}2.Definition 2.13Let us assume that associated with XXwe have a fundamental family (dx)x∈X{\left({d}_{x})}_{x\in X}of separation functions. We will say that this family is pseudo symmetric if there exists a function θd∈Θ{\theta }_{d}\in \Theta such that (2.9)dy(x)≤θd(dx(y)).{d}_{y}\left(x)\le {\theta }_{d}({d}_{x}(y)).Remark 2.14“Pseudo symmetry” is useful because it tells us that the topology is equivalently generated by the inverse balls B˜(x,ε)={y∈X:dy(x)<ε}.\tilde{B}\left(x,\varepsilon )=\{y\in X:{d}_{y}\left(x)\lt \varepsilon \}.This tells us that limn→∞xn=x⇔dx(xn)→n0⇔dxn(x)→n0.\mathop{\mathrm{lim}}\limits_{n\to \infty }{x}_{n}=x\hspace{0.33em}\iff \hspace{0.33em}{d}_{x}\left({x}_{n}){\to }_{n}0\hspace{0.33em}\iff \hspace{0.33em}{d}_{{x}_{n}}\left(x){\to }_{n}0.Compare Definition 3.4 in [5].2.5Set separation functionDefinition 2.15Mimicking the definition in metric space, we define, for A⊆XA\subseteq X, the set separation function(2.10)dA(y)=definfx∈Adx(y).{d}_{A}(y)\mathop{=}\limits^{{\rm{def}}}\mathop{\inf }\limits_{x\in A}{d}_{x}(y).Lemma 2.16dA(y){d}_{A}(y)is continuous.ProofWe know that dA(y){d}_{A}(y)is upper semicontinuous, so we need to prove that it is lower semicontinuous, that is, that V={y∈X:dA(y)>ε}V=\{y\in X:{d}_{A}(y)\gt \varepsilon \}is open, for ε≥0\varepsilon \ge 0. Let y∈Vy\in Vand let α=dA(y)>ε\alpha ={d}_{A}(y)\gt \varepsilon ; let then ε<β<α\varepsilon \lt \beta \lt \alpha and let r=ρd(y,α,β)r={\rho }_{d}(y,\alpha ,\beta ), where ρd{\rho }_{d}was defined in the pseudo reverse triangle inequality equation (2.3). We prove that B(y,r)⊆V;B(y,r)\subseteq V;indeed for any x∈Ax\in Awe have dx(y)≥α{d}_{x}(y)\ge \alpha : so, if dy(z)<r{d}_{y}\left(z)\lt r, then dx(z)>β{d}_{x}\left(z)\gt \beta and hence dA(z)≥β{d}_{A}\left(z)\ge \beta .□Remark 2.17Since {y∈X:dA(y)=0}⊇A\{y\in X:{d}_{A}(y)=0\}\supseteq Aand the left-hand side (LHS) is closed, in general we have {y∈X:dA(y)=0}⊇A¯.\{y\in X:{d}_{A}(y)=0\}\supseteq \overline{A}.To obtain equality we add pseudo symmetry.Hypothesis 2.18From here on we assume that we have a fundamental family (dx)x∈X{\left({d}_{x})}_{x\in X}of separation functions that is pseudo symmetric.Lemma 2.19Assuming that Hypothesis 2.18 holds then {y∈X:dA(y)=0}=A¯.\{y\in X:{d}_{A}(y)=0\}=\overline{A}.ProofWe need to prove that {y∈X:dA(y)=0}⊆A¯.\{y\in X:{d}_{A}(y)=0\}\subseteq \overline{A}.Let then dA(y)=0{d}_{A}(y)=0, which means that there is a sequence (xn)n⊆A{\left({x}_{n})}_{n}\subseteq Asuch that dxn(y)→n0{d}_{{x}_{n}}(y){\to }_{n}0. Then by pseudo symmetry we obtain y∈Ay\in A.□Corollary 2.20In particular, when Hypothesis 2.18 holds the topological space is T6, a.k.a. a perfectly normal Hausdorff space.2.6MetrizationWe have then a first metrization theorem, following the Urysohn metrization theorem.Theorem 2.21Assuming that Hypothesis 2.18 holds and the topological space (X,τ)\left(X,\tau )is second countable, then it is metrizable.ProofIndeed by Corollary 2.20 the space is T6.□2.7Diameter of pseudo ballsDefinition 2.22Let diam(A)=defsupx,y∈Adx(y){\rm{diam}}\left(A)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x,y\in A}{d}_{x}(y)be the diameter of a set A⊆XA\subseteq X.Lemma 2.23If Hypothesis 2.18 holds, then limε→0diam(B(y,ε))=0.\mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{\rm{diam}}\left(B(y,\varepsilon ))=0.If ρd{\rho }_{d}does not depend on yy, then the limit above is uniform in yy.ProofThe map ε↦diam(B(y,ε))\varepsilon \mapsto {\rm{diam}}\left(B(y,\varepsilon ))is monotonic. Let us fix α>0\alpha \gt 0and β=α/2\beta =\alpha \hspace{0.1em}\text{/}\hspace{0.1em}2; we will use the pseudo triangle inequality that is equation (2.2). There is ε>0\varepsilon \gt 0small enough such that θd(ε)<β{\theta }_{d}\left(\varepsilon )\lt \beta and ε<ρd(y,α,β)\varepsilon \lt {\rho }_{d}(y,\alpha ,\beta ). Choose any x,z∈B(y,ε)x,z\in B(y,\varepsilon ). By pseudo symmetry x∈B(y,ε)⇔dy(x)<ε⇒dx(y)≤θd(ε)<β,x\in B(y,\varepsilon )\hspace{0.33em}\iff \hspace{0.33em}{d}_{y}\left(x)\lt \varepsilon \Rightarrow {d}_{x}(y)\le {\theta }_{d}\left(\varepsilon )\lt \beta ,so by (2.2) dx(z)<α{d}_{x}\left(z)\lt \alpha .□2.8Uniform modulus“Uniform modulus” is the case when the modulus ρd{\rho }_{d}does not depend on yy. We saw in Lemma 2.23 that in this case we obtain stronger results. In the following, we discuss further results that use uniformity and pseudo symmetry.Hypothesis 2.24From here on we assume that we have a fundamental family d=(dx)x∈Xd={\left({d}_{x})}_{x\in X}of separation functions that is pseudo symmetric, and where the modulus ρd{\rho }_{d}does not depend on yy.We recall that the topology τ\tau was defined in Theorem 2.3.Lemma 2.25Assuming that Hypothesis 2.24 holds, then the topological space (X,τ)\left(X,\tau )is separable if and only if it is second countable.ProofOne implication is well known.We recall that the family of B(x,α)B\left(x,\alpha ), for x∈Xx\in Xand α>0\alpha \gt 0, is a global base for the topology τ\tau , see Theorem 2.3. Let (xn)n∈N{\left({x}_{n})}_{n\in {\mathbb{N}}}be a dense subset of XX, we prove that B(xn,1/m)B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m)is a global base, for n,m≥1n,m\ge 1integers; to this end for any B(x,α)B\left(x,\alpha )we will prove that there are n,mn,msuch that x∈B(xn,1/m)⊆B(x,α).x\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m)\subseteq B\left(x,\alpha ).Indeed, we choose mmsuch that (2.11)diamB(y,1/m)<α{\rm{diam}}B(y,1\hspace{0.1em}\text{/}\hspace{0.1em}m)\lt \alpha for any yy. Then, by pseudo symmetry, there is an nnsuch that dxn(x)<1/m{d}_{{x}_{n}}\left(x)\lt 1\hspace{0.1em}\text{/}\hspace{0.1em}m: this implies that x∈B(xn,1/m)x\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m). Then, by (2.11), for any z∈B(xn,1/m)z\in B\left({x}_{n},1\hspace{0.1em}\text{/}\hspace{0.1em}m), we have dx(z)<α{d}_{x}\left(z)\lt \alpha , so we conclude that z∈B(x,α)z\in B\left(x,\alpha ).□2.9Forward and reverseProposition 2.26If Hypothesis 2.24 holds, then the “pseudo triangle inequality” (2.2) and the “pseudo reverse triangle inequality” (2.3) are equivalent.ProofWe prove that if (2.2) holds then (2.3) holds. (The other implication is similar.) Consider the formula (2.12)dx(z)≥α∧dz(y)<ρ˜d⇒dx(y)>β{d}_{x}\left(z)\ge \alpha \wedge {d}_{z}(y)\lt {\tilde{\rho }}_{d}\Rightarrow {d}_{x}(y)\gt \beta that is equivalent to (2.3), since it was obtained by switching y,zy,zand replacing the modulus. We prove that for 0<β<α0\lt \beta \lt \alpha there exists ρ˜d=ρ˜d(α,β)>0{\tilde{\rho }}_{d}={\tilde{\rho }}_{d}\left(\alpha ,\beta )\gt 0such that ∀x,y,z∈X\forall x,y,z\in X, (2.12) holds.The condition (2.2) equivalently tells us that dx(y)>β∨dy(z)≥ρd∨dx(z)<α.{d}_{x}(y)\gt \beta \vee {d}_{y}\left(z)\ge {\rho }_{d}\vee {d}_{x}\left(z)\lt \alpha .If dx(z)≥α∧dz(y)<θ−1(ρd),{d}_{x}\left(z)\ge \alpha \wedge {d}_{z}(y)\lt {\theta }^{-1}\left({\rho }_{d}),then (by pseudo symmetry (2.9)) dy(z)≤θd(dz(y))<ρd;{d}_{y}\left(z)\le {\theta }_{d}({d}_{z}(y))\lt {\rho }_{d};hence dx(y)>β{d}_{x}(y)\gt \beta . We then define ρ˜d=θd−1∘ρd{\tilde{\rho }}_{d}={\theta }_{d}^{-1}\circ {\rho }_{d}with the convention in Definition 2.11.□2.10Delta complementWe extract a lemma from the proof of the following Theorem 2.29.Definition 2.27We assume Hypothesis 2.18. Given C⊂XC\subset Xclosed and δ>0\delta \gt 0we define Fδ(C)={x∈X:dC(x)>δ};{F}_{\delta }\left(C)=\left\{x\in X:{d}_{C}\left(x)\gt \delta \right\};we call Fδ(C){F}_{\delta }\left(C)the “delta-complemented set” since Fδ(C)∩C=∅,{F}_{\delta }\left(C)\cap C=\varnothing ,Fδ(C){F}_{\delta }\left(C)is open, and ⋃δ>0Fδ(C)=X⧹C.\bigcup _{\delta \gt 0}{F}_{\delta }\left(C)=X\setminus C.(All this follows from Lemmas 2.16 and 2.19; Figure 2.) Note also that for 0<s<t0\lt s\lt twe have Fs(C)⊇Ft(C),{F}_{s}\left(C)\supseteq {F}_{t}\left(C),while for C1⊆C2{C}_{1}\subseteq {C}_{2}closed sets we have (2.13)Fs(C1)⊇Fs(C2).{F}_{s}\left({C}_{1})\supseteq {F}_{s}\left({C}_{2}).Figure 2δ\delta complement.Lemma 2.28We assume Hypothesis 2.24. Let now s,t∈Rs,t\in {\mathbb{R}}with 0<s<t0\lt s\lt t: there exists an ε>0\varepsilon \gt 0such that, for any w∈Xw\in X, for any C⊆XC\subseteq Xclosed, if B(w,ε)∩Ft(C)≠∅,B\left(w,\varepsilon )\cap {F}_{t}\left(C)\ne \varnothing ,then B(w,ε)⊆Fs(C).B\left(w,\varepsilon )\subseteq {F}_{s}\left(C).ProofLet 0<s<s˜<t0\lt s\lt \tilde{s}\lt t. By the pseudo reverse triangle inequality (2.3), let ρ>0\rho \gt 0be small enough so that dx(z)≥t∧dz(y)<ρ⇒dx(y)>s˜.{d}_{x}\left(z)\ge t\wedge {d}_{z}(y)\lt \rho \Rightarrow {d}_{x}(y)\gt \tilde{s}.Then choose ε>0\varepsilon \gt 0small enough so that, by Lemma 2.23, we have ∀w∈X,∀z,y∈B(w,ε)⇒dz(y)<ρ.\forall w\in X,\forall z,y\in B\left(w,\varepsilon )\Rightarrow {d}_{z}(y)\lt \rho .Let us now suppose that z∈B(w,ε)∩Ft(C).z\in B\left(w,\varepsilon )\cap {F}_{t}\left(C).For all x∈Cx\in C, we have dx(z)>t{d}_{x}\left(z)\gt t. Consider any y∈B(w,ε)y\in B\left(w,\varepsilon ), then we have dz(y)<ρ{d}_{z}(y)\lt \rho . We conclude that dx(y)>s˜{d}_{x}(y)\gt \tilde{s}, hence infx∈Cdx(y)≥s˜{\inf }_{x\in C}{d}_{x}(y)\ge \tilde{s}. We have so proved that y∈Fs(A).y\in {F}_{s}\left(A).□2.11MetrizabilityTheorem 2.29We assume Hypothesis 2.24. If A{\mathcal{A}}is an open covering of X, then there exists ℰ{\mathcal{ {\mathcal E} }}an open covering of X that is countably locally finite, and ℰ{\mathcal{ {\mathcal E} }}is a refinement of A{\mathcal{A}}.(The proof is in Section A.2 on page 91.)Theorem 2.30If we assume Hypothesis 2.24, then the topological space (X,τ)\left(X,\tau )is metrizable.ProofThis follows from the above theorem, Corollary 2.20, and the Nagata-Smirnov metrization theorem (Sections 6-2 and 6-3 in [8]).□2.12Example: topological manifoldsAs an example, we propose this construction. In this section, we consider a topological manifold (X,τ)\left(X,\tau )that is Hausdorff, second countable, and locally Euclidean with dimension mm. (See §36 in [8] for further details.) Then (X,τ)\left(X,\tau )is paracompact and σ\sigma -compact. So there exists an atlas of homeomorphisms φi:Vi→Rm{\varphi }_{i}:{V}_{i}\to {{\mathbb{R}}}^{m}satisfying these additional conditions: For this atlas we have that the index family is I=NI={\mathbb{N}}, or IIis finite;For each i∈Ii\in Iwe have that Vi⊆X{V}_{i}\subseteq Xis open with compact closure V¯i{\overline{V}}_{i};Moreover, (Vi)i∈I{\left({V}_{i})}_{i\in I}is a locally finite open cover of XX.Theorem 2.31For x,y∈Xx,y\in Xwe can define(2.14)dx(y)=min{∣φi(x)−φi(y)∣Rm:i∈I∧x∈Vi∧y∈Vi}.{d}_{x}(y)=\min \left\{| {\varphi }_{i}\left(x)-{\varphi }_{i}(y){| }_{{{\mathbb{R}}}^{m}}:i\in I\wedge x\in {V}_{i}\wedge y\in {V}_{i}\right\}.Note that the set on the RHS is finite; if it is empty, then we set dx(y)=+∞{d}_{x}(y)=+\infty . These functions dx(y){d}_{x}(y)are continuous (jointly in x,yx,y) and are a fundamental family of separation functions that generates the topology τ\tau of XX.(Although the above result seems to be intuitive, the proof is surprisingly long and intricate, so it was moved to Section A.3.)Interestingly, the modulus ρd{\rho }_{d}associated with the family dx(y){d}_{x}(y)can be defined to satisfy the requirements of Proposition 2.10.Note that dx(y)=dy(x){d}_{x}(y)={d}_{y}\left(x), so this family satisfies Hypothesis 2.18. Consequently, the set separation function dA(y){d}_{A}(y)satisfies Lemmas 2.16 and 2.19: this explicitly proves that the space is T6. So Theorems 2.31 and 2.21 are an alternative way to prove that a manifold (X,τ)\left(X,\tau )is metrizable when it satisfies the hypotheses listed at the beginning of this section.Example 2.32In general, the function dx(y){d}_{x}(y)defined above in (2.14) does not satisfy the triangle inequality. Consider this example of a manifold covered with two charts, where x∈V1⧹V2,y∈V1∩V2,z∈V2⧹V1x\in {V}_{1}\setminus {V}_{2},\hspace{1.0em}y\in {V}_{1}\cap {V}_{2},\hspace{1.0em}z\in {V}_{2}\setminus {V}_{1}and dx(y)<∞,dy(z)<∞,dx(z)=∞.{d}_{x}(y)\lt \infty ,\hspace{1.0em}{d}_{y}\left(z)\lt \infty ,\hspace{1.0em}{d}_{x}\left(z)=\infty .Example 2.33Consider X=RX={\mathbb{R}}and cover it with charts having Vn=(n−1,n+1){V}_{n}=\left(n-1,n+1)and φn(x)=1max{1,∣n∣}ψ(x−n),ψ(x)=x(1−x2){\varphi }_{n}\left(x)=\frac{1}{\max \left\{1,| n| \right\}}\psi \left(x-n),\hspace{1.0em}\psi \left(x)=\frac{x}{\left(1-{x}^{2})}for n∈Zn\in {\mathbb{Z}}. Let then x=n,y=n+12,z=n+1,x=n,\hspace{1em}y=n+\frac{1}{2},\hspace{1em}z=n+1,so dx(y)=dy(z)=23n,dx(z)=∞.{d}_{x}(y)={d}_{y}\left(z)=\frac{2}{3n},\hspace{1.0em}{d}_{x}\left(z)=\infty .This explains the importance of the dependence of ρd{\rho }_{d}on yy.In some cases, it may happen that we do not know an easy formula for the distance that metrizes the manifold XX.The metrization theorems, such as Nagata-Smirnov metrization theorem and Urysohn’s metrization theorem, are usually proven by showing that there is an embedding of XXinto RN{{\mathbb{R}}}^{{\mathbb{N}}}; to define this embedding, Urysohn’s lemma is exploited to define countably many functions fn:X→[0,1]{f}_{n}:X\to \left[0,1]; then a distance is defined on RN{{\mathbb{R}}}^{{\mathbb{N}}}and pulled back on XX. While perfectly valid as a proof, it is not an easily manageable definition and it is unsuitable for numerical algorithms.If XXis compact, then XXcan be embedded in RN{{\mathbb{R}}}^{N}; so this can be used to define a distance on XX, by carefully tracking how the embedding is defined (as e.g. in §36 in [8]). This plan could be carried on, eventually providing an explicit formula for the distance; in particular, we can assume that the atlas is finite, let #I\#Ibe its cardinality, then such proof provides N=(m+1)#IN=\left(m+1)\#I.We remark that, at the same time, formula (2.14) gives us a very convenient definition of separation functions (also when XXis not compact): those encode the idea of “nearness” and can be used in further proofs and/or for numerical algorithms.3Mild topologyIn this section, we propose a novel topology on the space of continuous functions C0(M;N){C}^{0}\left(M;\hspace{0.33em}N).To define the mild topology we need that (N,dN)\left(N,{d}_{N})be a metric space.We fix a distinguished point p¯∈N\overline{p}\in N.Let f,g∈C0=C0(M;N)f,g\in {C}^{0}={C}^{0}\left(M;\hspace{0.33em}N), we define the “mild separation” (3.1)dfmild,p¯(g)=defsupx∈MdN(f(x),g(x))1+dN(f(x),p¯).{d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\frac{{d}_{N}(f\left(x),g\left(x))}{1+{d}_{N}(f\left(x),\overline{p})}.For f∈C0f\in {C}^{0}and α>0\alpha \gt 0we define the “mild pseudo ball” (3.2)Bmild,p¯(f,α)=def{g∈C0:df(g)<α}.{B}^{\text{mild},\overline{p}}(f,\alpha )\mathop{=}\limits^{{\rm{def}}}\left\{g\in {C}^{0}:{d}_{f}\left(g)\lt \alpha \right\}.We omit the superscripts “mild,p¯\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}” for ease of notation.Definition 3.1The mild topology on C0{C}^{0}is the topology generated by the above sets B(f,α)B(f,\alpha ). We will write CM0(M;N){C}_{M}^{0}\left(M;\hspace{0.33em}N)to denote this topological space.The above definitions will be justified in Proposition 3.5. To this end, we prove these two lemmas as follows.Lemma 3.2(Pseudo symmetry) dg(f)≤θ(df(g)){d}_{g}(f)\le \theta \left({d}_{f}\left(g))with θ(α)=α1−αα<1∞α≥1.\theta \left(\alpha )=\left\{\begin{array}{ll}\frac{\alpha }{1-\alpha }\hspace{1.0em}& \alpha \lt 1\\ \infty \hspace{1.0em}& \alpha \ge 1.\end{array}\right.ProofSuppose 0<α<10\lt \alpha \lt 1and df(g)≤α{d}_{f}\left(g)\le \alpha then dN(f(x),g(x))≤α(1+dN(f(x),p¯))≤α(1+dN(g(x),p¯)+dN(g(x),f(x))),{d}_{N}(f\left(x),g\left(x))\le \alpha (1+{d}_{N}(f\left(x),\overline{p}))\le \alpha (1+{d}_{N}(g\left(x),\overline{p})+{d}_{N}(g\left(x),f\left(x))),hence □(1−α)dN(f(x),g(x))≤α(1+dN(g(x),p¯)).\left(1-\alpha ){d}_{N}(f\left(x),g\left(x))\le \alpha (1+{d}_{N}(g\left(x),\overline{p})).Lemma 3.3(Pseudo triangle inequality) Let f,g,h∈C0f,g,h\in {C}^{0}and α>0\alpha \gt 0; if df(g)≤β<α{d}_{f}\left(g)\le \beta \lt \alpha and dg(h)≤ρd(α,β){d}_{g}\left(h)\le {\rho }_{d}\left(\alpha ,\beta )with(3.3)ρd(α,β)=α−β1+β,{\rho }_{d}\left(\alpha ,\beta )=\frac{\alpha -\beta }{1+\beta },then df(h)≤α{d}_{f}\left(h)\le \alpha .Proofdf(g)≤β{d}_{f}\left(g)\le \beta means dN(f(x),g(x))≤β(1+dN(f(x),p¯)),{d}_{N}(f\left(x),g\left(x))\le \beta (1+{d}_{N}(f\left(x),\overline{p})),moreover, dg(h)≤ρ{d}_{g}\left(h)\le \rho means dN(g(x),h(x))≤ρ(1+dN(g(x),p¯));{d}_{N}\left(g\left(x),h\left(x))\le \rho (1+{d}_{N}\left(g\left(x),\overline{p}));summing them(3.4)dN(f(x),h(x))≤dN(f(x),g(x))+dN(g(x),h(x))≤(β+ρ)+βdN(f(x),p¯)+ρdN(g(x),p¯).{d}_{N}(f\left(x),h\left(x))\le {d}_{N}(f\left(x),g\left(x))+{d}_{N}\left(g\left(x),h\left(x))\le \left(\beta +\rho )+\beta {d}_{N}(f\left(x),\overline{p})+\rho {d}_{N}\left(g\left(x),\overline{p}).At the same time, dN(g(x),p¯)≤dN(g(x),f(x))+dN(f(x),p¯)≤β+(1+β)dN(f(x),p¯):{d}_{N}\left(g\left(x),\overline{p})\le {d}_{N}\left(g\left(x),f\left(x))+{d}_{N}(f\left(x),\overline{p})\le \beta +\left(1+\beta ){d}_{N}(f\left(x),\overline{p}):substituting this in (3.4), dN(f(x),h(x))≤(β+ρ(1+β))+(β+ρ(1+β))dN(f(x),p¯).{d}_{N}(f\left(x),h\left(x))\le (\beta +\rho (1+\beta ))+(\beta +\rho (1+\beta )){d}_{N}(f\left(x),\overline{p}).We just need to find a ρ>0\rho \gt 0such that (β+ρ(1+β))≤α:(\beta +\rho (1+\beta ))\le \alpha :the value defined in equation (3.3) is such a choice.□Remark 3.4Lemma 3.3 proves the pseudo triangle inequality in the form in equation (2.5); then the form in equation (2.2) follows from Proposition 2.10. Moreover, the pseudo reverse triangle inequality (2.3) holds as well, due to Lemma 3.2 and Proposition 2.26. So the family df{d}_{f}defined in formula (3.1) is indeed a family of separation functions, as defined in Definition 2.1.Summarizing, we can state the needed result.Proposition 3.5The pseudo balls B(f,α)B(f,\alpha )are a global base for the mild topology. The mild topology is metrizable.ProofThe first statement follows from Theorem 2.3. The second statement derives from Theorem 2.30; we need to verify the two Hypothesis 2.24. For the first hypothesis, we note that Lemma 3.2 proves that the family of separation functions is pseudo symmetric.The second hypothesis requires that the modulus ρ(y,α,β)\rho (y,\alpha ,\beta )appearing in Definition 2.1 does not depend on yy; this is satisfied by the modulus defined in equation (3.3).□Lemma 3.6The mild topology does not depend on the choice of p¯∈N\overline{p}\in N.ProofGiven p¯,p˜∈N\overline{p},\tilde{p}\in N, for any β>0\beta \gt 0, choosing α=β11+dN(p¯,p˜),\alpha =\beta \frac{1}{1+{d}_{N}\left(\overline{p},\tilde{p})},we have α(1+dN(y,p¯))≤α(1+dN(y,p˜)+dN(p˜,p¯))≤β(1+dN(y,p˜));\alpha \left(1+{d}_{N}(y,\overline{p}))\le \alpha \left(1+{d}_{N}(y,\tilde{p})+{d}_{N}\left(\tilde{p},\overline{p}))\le \beta \left(1+{d}_{N}(y,\tilde{p}));then we reason as in Remark 3.8.□Proposition 3.7The mild topology is stronger than the weak topology; and it is weaker than the strong topology.ProofWe show that the mild topology is stronger than the weak topology. Fix ε>0\varepsilon \gt 0and a compact set K⊆MK\subseteq M, let β=maxx∈KdN(f(x),p¯)\beta =\mathop{\max }\limits_{x\in K}{d}_{N}(f\left(x),\overline{p})and ρ<ε1+β,\rho \lt \frac{\varepsilon }{1+\beta },we know that if g∈B(f,ρ),g\in B(f,\rho ),then ∀x∈K,dN(f(x),g(x))<ε.\forall \hspace{-0.3em}x\in K,{d}_{N}(f\left(x),g\left(x))\lt \varepsilon .The fact that the mild topology is weaker than the strong topology follows from Remark 3.8.□Remark 3.8We may also define the “mild neighborhood” (3.5)B˜(f,α)=def{g∈C0:∀x∈M,dN(f(x),g(x))<α(1+dN(f(x),p¯))}.\tilde{B}(f,\alpha )\mathop{=}\limits^{{\rm{def}}}\left\{g\in {C}^{0}:\forall x\in M,{d}_{N}(f\left(x),g\left(x))\lt \alpha (1+{d}_{N}(f\left(x),\overline{p}))\right\}.Note that, for 0<β<α0\lt \beta \lt \alpha (3.6)B˜(f,β)⊆B(f,α)⊆B˜(f,α),\tilde{B}(f,\beta )\subseteq B(f,\alpha )\subseteq \tilde{B}(f,\alpha ),so “mild neighborhoods” can be used to define the mild topology; unfortunately, they may fail to be open.A “mild neighborhood” can be built using the same method seen in the graph topology (see Definition 1.2): indeed consider open sets of the form U={(x,y)∈M×N:∀x∈M,dN(f(x),y)<α(1+dN(f(x),p¯))}U=\left\{\left(x,y)\in M\times N:\forall x\in M,{d}_{N}(f\left(x),y)\lt \alpha (1+{d}_{N}(f\left(x),\overline{p}))\right\}for f∈C0,α>0f\in {C}^{0},\alpha \gt 0, and then B˜(f,α)={g∈C0:graph(g)∈U}.\tilde{B}(f,\alpha )=\left\{g\in {C}^{0}:\hspace{0.1em}\text{graph}\hspace{0.1em}\left(g)\in U\right\}.Consequently, equation (3.6) proves that the mild topology is coarser than the strong topology.Remark 3.9In general, this topology is not separable. For example, when N=M=RN=M={\mathbb{R}}, setting fs(x)=esx{f}_{s}\left(x)={e}^{sx}, we have dfs(ft)=1s>t∞s<t;{d}_{{f}_{s}}({f}_{t})=\left\{\begin{array}{ll}1\hspace{1.0em}& s\gt t\\ \infty \hspace{1.0em}& s\lt t;\end{array}\right.in these cases the topology does not satisfy the second countability axiom. (This is why we proved Theorem 2.30, that is based on Nagata-Smirnov metrization theorem; we cannot use Urysohn’s metrization theorem to prove that the mild topology is metrizable.)3.1MetrizabilityWe know by Lemma 3.2 and Theorem 2.30 that the mild topology is metrizable.At first sight, a reasonable candidate for a distance that generates the mild topology may be dmild?(f,g)=defsupx∈MdN(f(x),g(x))1+dN(f(x),p¯)+dN(g(x),p¯),{d}_{\text{mild?}}(f,g)\mathop{=}\limits^{{\rm{def}}}\mathop{\sup }\limits_{x\in M}\frac{{d}_{N}(f\left(x),g\left(x))}{1+{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),\overline{p})},where f,g∈C0(M;N)f,g\in {C}^{0}\left(M;\hspace{0.33em}N). Note that 0≤dmild?(f,g)<10\le {d}_{\text{mild?}}(f,g)\lt 1.Lemma 3.10Obviously dmild?(f,g)≤dfmild,p¯(g);{d}_{\text{mild?}}(f,g)\le {d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g);indeed the formula defining the LHS has one more positive term in the denominator than the formula defining the RHS. Moreover, for 0<α<10\lt \alpha \lt 1α=dmild?(f,g)⇒dfmild,p¯(g)≤2α1−α.\alpha ={d}_{\text{mild?}}(f,g)\Rightarrow {d}_{f}^{\hspace{0.1em}\text{mild}\hspace{0.1em},\overline{p}}\left(g)\le \frac{2\alpha }{1-\alpha }.ProofIf 0<α<10\lt \alpha \lt 1and dmild?(f,g)≤α{d}_{\text{mild?}}(f,g)\le \alpha , then dN((fx),g(x))≤α(1+dN(f(x),p¯)+dN(g(x),p¯))≤α(1+2dN(f(x),p¯)+dN(g(x),f(x)));{d}_{N}\left((fx),g\left(x))\le \alpha (1+{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),\overline{p}))\le \alpha (1+2{d}_{N}(f\left(x),\overline{p})+{d}_{N}(g\left(x),f\left(x)));hence □(1−α)dN(f(x),g(x))≤2α(1+dN(f(x),p¯)).\left(1-\alpha ){d}_{N}(f\left(x),g\left(x))\le 2\alpha (1+{d}_{N}(f\left(x),\overline{p})).So if dmild?(f,g){d}_{\text{mild?}}(f,g)is a distance, it will generate the mild topology.But is it a distance? The formula is obviously symmetric, and we have dmild?(f,g)=0⇔f≡g;{d}_{\text{mild?}}(f,g)=0\hspace{0.33em}\iff \hspace{0.33em}f\equiv g;the question is as follows: does it satisfy the triangle inequality?Consider then this formula dN?(z,w)=defdN(z,w)1+dN(z,p¯)+dN(w,p¯){d}_{\text{N?}}\left(z,w)\mathop{=}\limits^{{\rm{def}}}\frac{{d}_{N}\left(z,w)}{1+{d}_{N}(z,\overline{p})+{d}_{N}(w,\overline{p})}for z,w∈Nz,w\in N; so dmild?(f,g)=supx∈MdN?(f(x),g(x)).{d}_{\text{mild?}}(f,g)=\mathop{\sup }\limits_{x\in M}{d}_{\text{N?}}(f\left(x),g\left(x)).We note that dmild?(f,g){d}_{\text{mild?}}(f,g)satisfies the triangle inequality if and only if dN?(z,w){d}_{\text{N?}}\left(z,w)does. (For one implication, consider constant functions; for the other, use standard properties of the supremum.)Unfortunately, the quantity dN?(z,w){d}_{\text{N?}}\left(z,w)does not satisfy the triangle inequality for some choices of NN; as is seen in this example: let NNbe a circle of length 13 where the points are posed as in Figure 3.Figure 3Points along a circle NNof length 13 and distances.Remark 3.11At the same time, consider the case when NNis a Hilbert space and p¯=0\overline{p}=0, then dN?(z,w)=def‖z−w‖N1+‖z‖N+‖w‖N.{d}_{\text{N?}}\left(z,w)\mathop{=}\limits^{{\rm{def}}}\frac{\Vert z-w{\Vert }_{N}}{1+\Vert z{\Vert }_{N}+\Vert w{\Vert }_{N}}.Note that the formula is invariant for rotations, so it is enough to check the triangle inequality for N=R3N={{\mathbb{R}}}^{3}; numerical experiments suggest that it is indeed a distance; to this end, we tested the triangle inequality with randomly sampled points and tried to numerically minimize the difference dN?(x,y)+dN?(y,z)−dN?(x,z){d}_{\text{N?}}\left(x,y)+{d}_{\text{N?}}(y,z)-{d}_{\text{N?}}\left(x,z)for x,y,z∈R3x,y,z\in {{\mathbb{R}}}^{3}. We though could not prove it analytically. See addendum material for more information.3.2Properties of proper mapsLemma 3.12Suppose that (N,dN)\left(N,{d}_{N})is a “proper metric space,” i.e., closed balls are compact.If 0<α<10\lt \alpha \lt 1and f∈B(g,α)f\in B\left(g,\alpha ), then ggis proper iff f is proper.Similarly for the “mild neighborhood” B˜(f,α)\tilde{B}(f,\alpha )defined in Remark 3.8.ProofSuppose that dg(f)=D<∞{d}_{g}(f)=D\lt \infty and ffis proper, we prove that ggis proper. Let K⊆NK\subseteq Nbe compact, let R=maxy∈KdN(y,p¯),R=\mathop{\max }\limits_{y\in K}{d}_{N}(y,\overline{p}),and let H={y∈N:dN(y,p¯)≤D+R(D+1)};H=\{y\in N:{d}_{N}(y,\overline{p})\le D+R\left(D+1)\};then HHis compact. We prove that g−1(K)⊆f−1(H),{g}^{-1}\left(K)\subseteq {f}^{-1}\left(H),so that g−1(K){g}^{-1}\left(K)is compact. Indeed, if x∈g−1(K)x\in {g}^{-1}\left(K), then g(x)∈Kg\left(x)\in Kso dN(g(x),p¯)≤R{d}_{N}\left(g\left(x),\overline{p})\le R, hence dN(f(x),p¯)≤dN(g(x),p¯)+dN(g(x),f(x))≤D+(D+1)dN(g(x),p¯)≤D+R(D+1),{d}_{N}(f\left(x),\overline{p})\le {d}_{N}\left(g\left(x),\overline{p})+{d}_{N}\left(g\left(x),f\left(x))\le D+\left(D+1){d}_{N}\left(g\left(x),\overline{p})\le D+R\left(D+1),so f(x)∈Hf\left(x)\in H. Suppose now that ggis proper and dg(f)<1{d}_{g}(f)\lt 1, then, by pseudo symmetry Lemma 3.2, df(g)<∞{d}_{f}\left(g)\lt \infty . So ffis proper.□Corollary 3.13The set of proper maps is both open and closed in the mild topology.3.3Properties of affine actionsTheorem 3.14Suppose that N=RnN={{\mathbb{R}}}^{n}, dN(x,y)=∣y−x∣{d}_{N}\left(x,y)=| y-x| is the usual Euclidean distance; endow C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})with the mild topology; then the actions listed in Section 1.5 are (jointly) continuous.(The proof is in page 23 in Section A.4.)Remark 3.15For some specific actions some extra information may be useful. Rotation. If we choose p¯=0\overline{p}=0for convenience in the definition equation (3.2) (as is made possible by Lemma 3.6), then, given a rotation R∈O(n)R\in O\left(n), the map f∈C0↦Rf∈C0f\in {C}^{0}\mapsto Rf\in {C}^{0}is an “isometry”: indeed, B(Rf,α)=RB(f,α)B\left(Rf,\alpha )=RB(f,\alpha )because dRfRg=dfg.{d}_{Rf}Rg={d}_{f}g.We also note that for S,R∈O(n)S,R\in O\left(n), ∣Rg(x)−Sg(x)∣≤‖R−S‖∣g(x)∣;| Rg\left(x)-Sg\left(x)| \le \Vert R-S\Vert | g\left(x)| ;so dRg(Sg)≤‖R−S‖{d}_{Rg}\left(Sg)\le \Vert R-S\Vert where ‖R−S‖\Vert R-S\Vert is a matrix (operator) norm.Rescaling. Let s>0s\gt 0, let m=min{1,s},M=max{1,s}m=\min \left\{1,s\right\},M=\max \left\{1,s\right\}then mdf(g)≤dsf(sg)≤Mdf(g),m{d}_{f}\left(g)\le {d}_{sf}\left(sg)\le M{d}_{f}\left(g),so f∈C0↦sf∈C0f\in {C}^{0}\mapsto sf\in {C}^{0}is again a homeomorphism. For the action s∈R↦sf∈C0s\in {\mathbb{R}}\mapsto sf\in {C}^{0}similarly ∣t−s∣m≤dsf(tf)≤∣t−s∣M.| t-s| m\le {d}_{sf}\left(tf)\le | t-s| M.3.4CaveatsWe have then seen many good properties of the mild topology; there are some drawbacks though. The mild topology depends on the choice of distance dN{d}_{N}.It is not invariant w.r.t. homeomorphisms as in Proposition 1.4; it is invariant only for right action, i.e., if ΦM:M˜→M{\Phi }_{M}:\tilde{M}\to Mis a homeomorphism, then the map f↦f∘ΦMf\mapsto f\circ {\Phi }_{M}is a homeomorphism between C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)and C0(M˜;N){C}^{0}\left(\tilde{M};\hspace{0.33em}N), where both spaces are endowed with the mild topology.The space C0(M;N){C}^{0}\left(M;\hspace{0.33em}N)with the mild topology may fail to be connected, since proper maps are open and closed.When N=RnN={{\mathbb{R}}}^{n}with Euclidean structure, the space C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n})with the mild topology is not in general a topological vector space; there are many reasons, we list some. For g∈C0g\in {C}^{0}fixed, the map f∈C0↦g+f∈C0f\in {C}^{0}\mapsto g+f\in {C}^{0}may fail to be continuous. For example, consider g(x)=−exg\left(x)=-{e}^{x}and C0=C0(R;R){C}^{0}={C}^{0}\left({\mathbb{R}};{\mathbb{R}}), f(x)=exf\left(x)={e}^{x}. Then the counter image of the mild pseudo ball B(0,1)B\left(0,1)is {h(x)+ex:h∈C0,supx∈Rn∣h(x)∣<1},\left\{\phantom{\rule[-1.25em]{}{0ex}}h\left(x)+{e}^{x}:h\in {C}^{0},\mathop{\sup }\limits_{x\in {{\mathbb{R}}}^{n}}| h\left(x)| \lt 1\right\},and it does not contain any mild pseudo ball B(ex,ε)B\left({e}^{x},\varepsilon ).For f∈C0f\in {C}^{0}fixed, the map λ∈R↦λf∈C0\lambda \in {\mathbb{R}}\mapsto \lambda f\in {C}^{0}may fail to be continuous at λ=0\lambda =0(adapting the previous example).The space may not be connected.4ConclusionWe have discussed a novel method to define topologies, by separation functions; we have shown that, even when the topology happens to be metrizable, it may happen that the actual metric is not known and/or that the separation functions are more manageable than the metric that metrizes the topology.We have studied the mild topology C0(M;N){C}^{0}\left(M;\hspace{0.33em}N); it has some good properties: proper maps are a closed and open subset of C0(M;N){C}^{0}\left(M;\hspace{0.33em}N), as in the case of the strong topology; affine actions on N=RnN={{\mathbb{R}}}^{n}are continuous on C0(M;Rn){C}^{0}\left(M;\hspace{0.33em}{{\mathbb{R}}}^{n}), as in the case of the weak topology.It is possible to define similar concepts for Cr(M;N){C}^{r}\left(M;\hspace{0.33em}N), the space of rrtimes differentiable maps between two differentiable manifolds M,NM,N; similar properties hold and can be extended to other interesting classes of maps such as immersions, free immersions, submersions, embeddings, diffeomorphisms; this may be argument of a forthcoming paper.

Journal

Analysis and Geometry in Metric Spaces – de Gruyter

Published: Jan 1, 2023

Keywords: continuous functions; proper maps; strong topology; weak topology; metrization; quasi metrics; asymmetric metrics; topological manifolds; 54C10; 54C35; 54E35; 58D15; 58D19; 46T05; 46T10

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