Lipschitz functions on topometric spaces

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, normality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space $X$ from the lattice of continuous $1$-Lipschitz functions on $X$, in analogy with the recovery of a compact topological space $X$ from the structure of (real or complex) functions on $X$.


Introduction
Topometric spaces are spaces equipped both with a metric and a topology, which need not agree. To be precise, Definition 0.1. A topometric space is a triplet (X, T , d), where T is a topology and d a metric on X, satisfying: (i) The distance function d : X 2 → [0, ∞] is lower semi-continuous in the topology.
(ii) The metric refines the topology.
We follow the convention that unless explicitly qualified, the vocabulary of general topology (compact, continuous, etc.) refers to the topological structure, while the vocabulary of metric spaces (Lipschitz function, etc.) refers to the metric structure. Excluded from this convention are separation axioms: we assimilate the lower semi-continuity of the distance function to the Hausdorff separation axiom, and stronger axioms, such as normality and complete regularity, will be defined for topometric spaces below.
Compact topometric spaces were first defined in [BU10] as a formalism for various global and local type spaces arising in the context of continuous first order logic. General topometric spaces (i.e., non compact) were defined studied further from an abstract point of view in [Ben08b]. Further examples include types spaces for unbounded logic (merely locally compact), perturbation structures on type spaces. A very different class of examples, very far from being compact or even locally compact in general, is formed by automorphism groups of (metric) structures, as well as Polish groups or (completely) metrisable ones.
In addition, there are two classes of examples which recur throughout the paper, arising from the embedding of the categories of (Hausdorff) topological spaces and of metric spaces in the category of topometric spaces. By a maximal topometric space we mean one equipped with the discrete 0/1 distance, which can be identified, for all (or most) intents and purposes, with its underlying pure topological structure. Similarly, a minimal topometric space is one in which the metric and topology agree, which may be identified with its underlying metric structure. These sometimes serve as first sanity checks (e.g., when we define a normal topometric space we must check that a maximal one is normal if and only if it is normal as a pure topological space, and that minimal ones are always normal).
The aim of this paper is to study some basic properties of the class of (topologically) continuous and (metrically) Lipschitz functions on a topometric space. These are naturally linked with separation axioms. For example, existence results such as Urysohn's Lemma and Tietze's Extension Theorem are tied with normality, discussed in Section 1, while the Stone-Čech compactification (defined in terms of a universal property with respect to continuous Lipschitz functions) is related to complete regularity, as discussed in Section 2. To conclude, Section 3 characterises the bare minimum that the set of Lipschitz functions needs to satisfy.
Lipschitz functions on an ordinary metric spaces, and algebras thereof, are extensively studied in Weaver [Wea99]. This is some natural resemblance between our object of study here and that of Weaver, with the increased complexity due to the additional topological structure. Th reader may wish to compare, for example, our version of Tietze's Extension Theorem (Theorem 1.9) with [Wea99, Theorem 1.5.6] (as well as with the classical version of Tietze's Theorem, see Munkres [Mun75]).

Normal topometric spaces and Urysohn-Tietze results
For two topometric spaces X and Y we define C L(1) (X, Y ) to be the set of all continuous 1-Lipschitz functions from X to Y . An important special case is C L(1) (X) = C L(1) (X, C), where C is equipped with the standard metric and topology (i.e., with the standard minimal topometric structure), which codes information both about the topology and about the metric structure of X. In the present paper we seek conditions under which C L(1) (X) codes the entire topometric structure, as well as analogues of classical results related to separation axioms, in which C(X) would be replaced with C L(1) (X). As discussed in [Ben08b], we consider the lower semi-continuity of the distance function to be a topometric version of the Hausdorff separation axiom, so we may expect other classical separation axioms to take a different form in the topometric setting. We start with normality.
Definition 1.1. Let X be a topometric space. We say that a closed set F ⊆ X has closed metric neighbourhoods if for every r > 0 the set B(F, r) = {x ∈ X : d(x, F ) ≤ r} is closed in X.
We say that X admits closed metric neighbourhoods if all closed subsets of X do.
It was shown in [Ben08b] that compact sets always have closed metric neighbourhoods, so a compact topometric space admits closed metric neighbourhoods. Indeed, the first definition of a compact topometric space in [BU10] was given in terms of closed metric neighbourhoods. While this property seems too strong to be part of the definition of a non compact topometric space, it will play a crucial role in this section.
Definition 1.2. A normal topometric space is a topometric space X satisfying: (i) Every two closed subset F, G ⊆ X with positive distance d(F, G) > 0 can be separated by disjoint open sets. (ii) The space X admits closed metric neighbourhoods.
One checks that a maximal topometric space X (i.e., equipped with the discrete 0/1 distance) is normal if and only if it is so as a topological space. Similarly, a minimal topometric space (i.e., equipped with the metric topology) is always normal. Also, every compact topometric space is normal (since it admits closed metric neighbourhoods and the underlying topological space is normal).
We contend that our definition of a normal topometric space is the correct topometric analogue of the classical notion of a normal topological space. This will be supported by analogues of Urysohn's Lemma and of Tietze's Extension Theorem. The technical core of the proofs (and indeed, the only place where the definition of a normal topometric space is used) lies in the following Definition and Lemma. Definition 1.3. Let X be a topometric space, c > 0 a constant, S ⊆ R and Ξ S = {(F α , G α ) : α ∈ S} a sequence of pairs of closed sets F α , G α ⊆ X.
(i) We say that Ξ S is an approximation of a strictly c-Lipschitz partial continuous function on X, or simply a partial c-Lipschitz Proof. Since the partial approximation is finite it is also c ′ -Lipschitz for some c ′ < c. Define: By construction d(K, L) > 0 and both are closed as finite unions of closed sets. Since X is normal we can find disjoint open sets U ⊇ K and V ⊇ L. We claim that F ′ β = F β ∪ V c and G ′ β = G β ∪ U c will do. The first two items are trivially verified, so we only need to check the last one. So assume that α < β. We already know by hypothesis that We show similarly that if β < α then d(F ′ β , G α )c > α−β, and we are done.

1.4
Lemma 1.5. Let X be a normal topometric space, Ξ S = {(F α , G α ) : α ∈ S} a finite c-Lipschitz approximation. Then for every β ∈ R there is a c-Lipschitz approximation Ξ ′ S∪{β} ⊇ Ξ S . Proof. We may assume that β / ∈ S, and let F β = G β = ∅. Then (F α , G α ) : α ∈ S ∪ {β} is a partial c-Lipschitz approximation and Lemma 1.4 (with the same β) we obtain the required approximation Proposition 1.6. In a normal topometric space every finite approximation of a c-Lipschitz continuous function approximates such a function.
Proof. Let X be a normal topometric space, {(F α , G α ) : α ∈ S} a finite c-Lipschitz approximation. Since S is finite its convex hull is a compact interval I ⊆ R. Let T ⊆ I be a countable dense subset containing S. By repeated applications of Lemma 1.5 one can extend the given approximation into a c-Lipschitz The topometric analogue of Urysohn's Lemma is obtained as an easy corollary.
Corollary 1.7 (Urysohn's Lemma for topometric spaces). Let X be a normal topometric space, F, G ⊆ X closed sets, 0 < r < d(F, G). Then there exists a 1-Lipschitz continuous function f : X → [0, r] equal to 0 on F and to r on G.
Conversely, every topometric space in which this property holds is normal.
Assume now that the first property holds in X. Then closed sets of positive distance can be separated by a 1-Lipschitz continuous function, and therefore by open sets. Also, if F ⊆ X is closed and d(x, F ) > r then we may separate F and x by a 1-Lipschitz continuous function such that f ↾ F = 0 and f (x) > r.
Then {y : f (y) ≤ r} is a closed set containing B(F, r) but not x. If follows that B(F, r) is closed.
is a partial c-Lipschitz approximation on X, so we may apply Lemma 1.4 to each α ∈ S and obtain the required approximation.

1.8
Observe that the forced limit operator F lim : Moreover, for an arbitrary topometric space the following are equivalent: (i) X is a normal topometric space.
(ii) Tietze's Extension Theorem for topometric spaces (i.e., the statement above) holds in X.
(iii) The statement of Proposition 1.6 holds in X.
(iv) Urysohn's Lemma (the main assertion of Corollary 1.7) holds in X.
For the moreover part, we have seen that if X is normal then (ii)-(iv) hold. Conversely, each of (ii) and (iii) clearly implies (iv), and by Corollary 1.7 (iv) implies that X is normal.

1.9
This proof of Tiezte's theorem is fairly different from other the author managed to find in the literature. Indeed none of the more common proofs seems to be capable of preserving the Lipschitz condition.

Completely regular topometric spaces and Stone-Čech compactification
Let {X i : i ∈ I} be a family of topometric spaces. We equip the set i∈I X i with the product topology and the supremum metric d(x,ȳ) = sup{d(x i , y i ) : i ∈ I}. One verifies easily the result is indeed a topometric space which we call the product topometric structure.
In particular we obtain large compact topometric spaces of the form [0, ∞] I , and we claim that these are in some sense universal, meaning that every compact topometric space embeds in one of those.
Similarly, every bounded compact topometric (i.e., of finite diameter) can be embedded in [0, M ] I , and up to re-scaling in [0, 1] I . In fact we shall show that every completely regular topometric space embeds in such a space, obtaining a Stone-Čech compactification.
Say that a family of functions F ⊆ C X separates points from closed sets if for every closed set F ⊆ X and x ∈ X F , there is a function f ∈ F which is constant on F and takes some different value at x.
Fact 2.1. Let X be a Hausdorff topological space, F ⊆ C(X) a family separating points from closed sets. Then the map θ : Proof. This is fairly standard. First of all F separates points so θ is injective. To see that θ is continuous, it is enough to consider a sub-basic open set is open and π f is the projection on the f th coördinate. Then θ −1 (U ) = f −1 (V ) is open. In order to show that θ is a homeomorphism with its image it will be enough to show that for F ⊆ X closed and x / ∈ F there is a closed set F ′ ⊆ C F such that θ(F ) ⊆ F ′ and θ(x) / ∈ F ′ . Since F separates points from closed sets there is f ∈ F such that f ↾ F = t and f (x) = t. Then F ′ = {ȳ ∈ C F : y f = t} will do.

2.1
Definition 2.2. Let X be a topometric space. Say that a family of functions F ⊆ C L(1) (X) is sufficient if (i) It separates points and closed sets.
(Clearly, ≥ always holds.) A topometric space X is completely regular if C L(1) (X) is sufficient. This is clearly equivalent to C L(1) (X, R + ) being sufficient.
In view of Fact 2.1 we may say that a topometric space X is completely regular if C L(1) (X) captures both the topological structure and the metric structure of X.

Proposition 2.3.
(i) Every normal topometric space is completely regular. (ii) Every subspace of a completely regular space is completely regular. (iii) Let X be a maximal topometric space. Then it is topologically completely regular if and only if it is topometrically completely regular.
Proof. The first item follows from Corollary 1.7, keeping in mind that since the metric of a topometric space X refines its topology, if F ⊆ X is closed and x / ∈ F then d(x, F ) > 0. For the second item, assume that X is completely regular, Thus if x ∈ Y F then x ∈ X F , so there is a 1-Lipschitz continuous function separating F from x, and its restriction to Y is continuous and 1-Lipschitz as well. The same argument works for witnessing distances.
The last item follows from the fact that every function from a maximal topometric space to [0, 1] is 1-Lipschitz.

2.3
Proposition 2.4. Let X be a completely regular topometric space and let F = C L(1) (X, R + ). Then the map θ : X → (R + ) F from Fact 2.1 is a topometric embedding, i.e., an isometric homeomorphic embedding.
Proof. Immediate from the definitions.

2.4
Corollary 2.5. Every completely regular topometric space X (and thus in particular every normal or compact one) embeds in some power of [0, ∞]. If in addition X is bounded, say of diameter 1, then it embeds in a power of [0, 1].
Proof. We just have to show the last part. Indeed let θ : X → [0, ∞] I be any embedding. Define Then θ ′ is an embedding as well, and0 ∈ θ(X). If X is bounded of diameter 1 then θ(X) ⊆ [0, 1] I .

2.5
Theorem 2.6. A topometric space admits a compactification if and only if it is completely regular.
Proof. If X is completely regular then we can identify it with a subspace of [0, ∞] I , and then its closure there is a compactification. Conversely, assume X admits a compactificationX. ThenX is completely regular, whereby so is X.

2.6
Theorem 2.7. Let X be completely regular. Then it admits a compactification βX satisfying the following universal property: Every 1-Lipschitz continuous function f : X → [0, ∞] can be extended to such a function on βX (and the extension is unique).
Moreover, βX is unique up to a unique isomorphism (i.e., isometric homeomorphism) and satisfies the same universal property with any compact topometric space Y instead of [0, ∞].
The uniqueness of an object satisfying this universal property is now standard.
Definition 2.8. The compactification βX, if it exists (i.e., if X is complete regular) is called the Stone-Čech compactification of X.
Automorphism groups of metric structures probably form the most natural class of examples of non (locally) compact topometric spaces. They are easily checked to be completely regular. Proof. Since d u (f, g) = sup a∈M d(f a, ga), and for each a the function (f, g) → d(f a, ga) is continuous, d u is lower semi-continuous. Assume that d u (f, g) > r. Then there exists a ∈ M such that d(f a, ga) > r, and we may define θ(x) = d(f a, xa). Then θ is continuous and 1-Lipschitz (by definition of point-wise and uniform convergence). In addition, θ(f ) = 0 and θ(g) > r. Thus continuous 1-Lipschitz functions witness distances, and it follows that d u is lower semi-continuous. Now let U be a topological neighbourhood of f . Then there is a finite tupleā ∈ M n and ε > 0 such that U contains the set Uā ,fā,ε = {h : d(hā, fā) < ε}. Then the function ρ(x) = d(fā, xā) separates f from G U .
A similar reasoning applies to the case of an abstract group (acting on itself on the left). In fact, when G is completely metrisable then this case can be shown to be a special case of the first, and every metrisable group can be embedded in a completely metrisable one.

2.9
Question 2.10. Are automorphism groups of metric structures topometrically normal? In other words, do continuous 1-Lipschitz functions witness distance between closed sets?
Most topometric spaces one would encounter, such as compact ones (e.g., type spaces) or automorphism groups, are (metrically) complete. If X is an incomplete topometric space then the metric structure carries obviously over to the completionX, and it is legitimate to ask whether, or how, the topological structure carries there as well. Let us concentrate on the case where X is completely regular.
Definition 2.11. Let X be a completely regular topometric space. We equip its completionX with the least topology such that for every f ∈ C L(1) (X), the unique 1-Lipschitz extension of f tof :X → C is continuous. In other words, we define it so that the restriction map C L(1) (X) → C L(1) (X) is a bijection.
Lemma 2.12. Let X be a completely regular topometric space. Then so isX.
Proof. The Stone-Čech compactification βX is compact and therefore complete, and the canonical identification ofX with a subset of βX is homeomorphic.

2.12
The topometric structure we put onX is clearly the strongest possible regular one, and it is natural to ask whether it is unique. For a positive result in this direction, let us consider the following two conditions on a topometric space X: Clearly ( * ) implies ( * * ).
Proposition 2.13. Let X be a completely regular topometric space in which condition ( * * ) holds, and let X 0 ⊆ X be a metrically dense subspace. Then every f ∈ C L(1) (X 0 ) extends tof ∈ C L(1) (X).

2.13
Lemma 2.14. Condition ( * ) holds in every topometric space of the form [s i , r i ]. More generally, it holds in every minimal or maximal topometric space, and if it holds in each X i then it holds in X i . Similarly, if condition ( * * ) holds in each X i then it also holds in X i .

2.14
Lemma 2.15. Condition ( * ) holds in every topometric group. In fact, while we usually require that the distance in a topometric group be biïnvariant, here it is enough that it be invariant on one side.
Proof. Assume that the distance is left-invariant. Then one checks that B(U, r) = d(h,1)<r U h.

2.15
On the other hands, it is not difficult to construct even compact topometric spaces where the properties discussed in this section fail.
In pure C * -algebraic terms, we can express C(X × X) as the C * tensor product C(X) ⊗ C(X), and define δ : C(X) → C(X) ⊗ C(X) by δf = f ⊗ 1 − 1 ⊗ f , i.e., δf (x, y) = f (x) − f (y). Since a point in X × X corresponds to a maximal ideal in C(X) ⊗ C(X), we obtain that (vi) is further equivalent to (iv ′′ ) If f / ∈ A then there exists ε > 0 such that the family of all ε− . |δf −δg|, in the sense of continuous functional calculus, as g varies over A, generates a proper ideal in C(X) ⊗ C(X).
Theorem 3.4. Let X be a compact topological space, A ⊆ C(X). Then the following are equivalent: (i) The set A is an L(1)-set.
(ii) There is a topometric structure (X, d) on X such that A = L L(1) (X). In this case the metric d is unique and can be recovered by (1) Proof. Bottom to top is easy, and (1) follows from Urysohn's Lemma for normal topometric spaces and the fact that a compact topometric space is normal. Assume therefore that A is an L(1)-set, and let us define d by (1).
Clearly d is a pseudo-distance, and is lower semi-continuous being the supremum of continuous functions. Since A separates points from closed sets, d refines the topology, and in particular is a distance (rather than a pseudo-distance). Thus (X, d) is a topometric space, and we view it henceforth as such. It is then immediate from the construction that A ⊆ C L(1) (X). Finally, assume that f / ∈ A, and let x, y ∈ X and ε > 0 be such that if |f (x)−g(x)−f (y)+g(y)| < ε then g / ∈ A. Since A is closed under multiplication by complex scalar of absolute value ≤ 1, this is only possible if |f (x)− f (y)| ≥ |g(x)− g(y)|+ ε for all g ∈ A. It follows that |f (x) − f (y)| ≥ d(x, y) + ε, so f / ∈ C L(1) (X), as desired.

3.4
This is quite different from [Wea99, Theorem 4.3.2], which still seems to be the most closely analogous result therein.