A Universal Separable Diversity

The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points. We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.


Introduction
The concepts of homogeneity and universality pervade many areas of mathematics. They appear in particular when the point of view of mathematical logic is adopted. Consider the Fraïssé limit [3] of a class of finite structures with the almagamation property. For instance, the Rado (or random) graph [9] is the Fraïssé limit of the class of undirected finite graphs. This graph is universal for the class of countable graphs, and ultrahomogeneous in the sense that its isomorphic finite subgraphs are automorphic in the graph. The conjunction of these two properties makes the Rado graph unique up to isomorphism. This behaviour is entirely typical for Fraïssé limits.
For structures in the classical sense, countability is essential to ensure this uniqueness. However, we are mainly interested in the setting of a complete metric space X We say that a diversity (X, δ) is complete if its induced metric (X, d) is complete [8] and that a diversity is separable if its induced metric is separable.
Our main goal is to construct the diversity analog (U, δ U ) of the Urysohn metric space. It is determined uniquely by being universal for separable diversities, and ultrahomogeneous in the sense that isometric finite subdiversities are automorphic.
The construction follows the same approach as Katětov's construction of the Urysohn universal metric space [5]. Starting with any diversity (X, δ), we first consider the set of all one-point extensions of X which we denote by E(X). Since E(X) turns out to not be separable under the natural metric, we instead consider extensions with finite support, which provides a separable diversity E(X, ω) in which (X, δ) is naturally embedded. We repeat this procedure obtaining a nested sequence of separable diversities. The analogue of the Urysohn metric space is constructed as the completion of the direct limit of all these diversities. Finally we show that this complete separable diversity has the diversity analogue of Urysohn's extension property, and hence is universal and ultrahomogeneous.
For each k ≥ 1, let δ k be the function that sends (a 1 , . . . , a k ) to δ({a 1 , . . . , a k }). By Prop. 1 below we can view a diversity (X, δ k k∈N ) as a metric structure in the sense of [12]. The Urysohn diversity can presumably also be obtained in the general framework of Ben Yaacov [11]. The amount of work needed to show the hypotheses for the general construction are satisfied would be about the same, and indeed quite similar to what we will do in this paper. Proposition 1. Let (X, δ) be a diversity. For each n, the function δ n is 1-Lipschitz in each argument.
Proof. Consider varying the ith argument of δ k from x i to x ′ i . We know from the triangle inequality that

Background and Preliminaries
Recall from above that any diversity (X, δ) has an induced metric (X, d) where d(a, b) = δ({a, b}) for all a, b ∈ X. Conversely, given any metric space (X, d), consider the diversities that have (X, d) as an induced metric. Lower and upper bounds on the possible diversities that have (X, d) as the induced metric are provided by the diameter diversity and the Steiner diversity.
For any metric space (X, d), the corresponding diameter diversity (X, δ diam ) is defined by On the other hand, given a metric space (X, d), consider the weighted complete graph (X, E, w) where X is the set of vertices, E is the set of all unordered pairs of vertices, and w assigns weight d(a, b) to the edge (a, b). A tree T with vertices in X covers a finite set A ⊆ X if A is a subset of the vertices of T . The Steiner diversity (X, δ Steiner ) is defined by letting δ Steiner (A) be the infimum, over all trees that cover A, of the total weight of the tree.
The diameter diversity and the Steiner diversity of a metric space (X, d) are important in that for any other diversity (X, δ) that has (X, d) as an induced metric space we have for all finite A ⊆ X [2]. Also, these two extreme diversities can be thought of as simple in the sense that the values of δ diam and δ Steiner on finite subsets of X are determined purely by their values on pairs of points.
At the end of the paper, we will show that the diversity analogue of the Urysohn metric space is neither a diameter diversity nor a Steiner diversity of any metric space. In particular, it is neither the diameter diversity nor the the Steiner diversity of the Urysohn metric space, even though it has the Urysohn metric space as its induced metric space.

Analogue of Katětov functions
For a metric space (X, d), a Katětov function f : X → R describes a potential onepoint extension of X by a point z: a metric d on X ∪ {z} extending d is given by defining d(x, z) = f (x) for each x ∈ X. By [5] we have E(X) is the set of Katětov functions, which form a metric space with the sup distance d ∞ (f, g) = sup x |f (x) − g(x)|. Identifying x ∈ X with the function y → d(x, y) isometrically embeds X into E(X). Let (X, δ) be a diversity. We will define its extension E(X) by adapting Katětov's approach [5].
for all finite A ⊆ X. The point z may be in X.
As before, each admissible function on (X, δ) corresponds to a way of extending (X, δ) by one point z. We let E(X) be the set of all admissible functions on (X, δ). We provide the analogue of Eq. (1).
Lemma 3. f : P fin (X) → R is in E(X) if and only if f satisfies the following: Proof. ⇒: Suppose f is admissible, so δ is a diversity on X ∪ {z}, and f (A) = δ(A ∪ {z}) for all A ∈ P fin (X). Then δ({z}) = 0 implies property (i). Monotonicity of δ(A) implies f (A) = δ(A ∪ {z}) ≥ δ(A) = δ(A), which is property (ii). The triangle inequality (D2) for δ gives, for all C = ∅, which is property (iii). Finally, using the triangle inequality for δ again gives which is property (iv). ⇐: Suppose now that f satisfies the properties (i) through (iv). If f ({x}) = 0 for some x ∈ X, let z = x. Otherwise let z ∈ X. Define δ on X ∪ {z} by To show that δ is subadditive on intersecting sets we again have several cases. If z ∈ A and z ∈ B then Hence δ is subadditive on intersecting sets. Together with monotonicity this gives the triangle inequality for diversities.
Analogous to the metric d ∞ in Katětov's construction, we define a diversity function δ on E(X). The motivating idea for our choice of function is that since every admissible function f corresponds to extending a diversity by an additional point z, considering admissible functions f 1 , . . . , f k should require us to extend the diversity by points z 1 , . . . , z k simultaneously, giving a new diversity δ E defined on X ∪{z 1 , . . . , z k }. This diversity must coincide with δ on X, and also satisfy that Once we have fixed a choice of δ E given these constraints, we let One choice for δ that turns out to generalize from the metric case nicely is to let δ to be the minimum diversity satisfying the constraints for all finite A ⊆ X. We now describe how to obtain an explicit expression for δ. We say that a collection of finite subsets E 1 , . . . , E k is connected if, when we partition E 1 , . . . , E k into two non-empty collections of sets, then there is an E i on one side of the partition and an E j on the other side of the partition such that E i ∩E j = ∅. Equivalently, define a graph with v 1 , . . . , v k corresponding to E 1 , . . . , E k and there is an edge between v i and v j if and only if E i ∩ E j = ∅. Then the collection of sets is connected iff the graph is connected.
Putting this into terms of admissible functions we get This puts the following lower bound on δ: Now this bound must hold for each choice of j and A i for i = j. This suggests the following definition of δ on E(X): where all A i are finite subsets of X. We define δ(∅) and δ({f }) to be zero, for all f ∈ E(X). Theorem 4 below shows that this is a diversity on E(X) that extends (X, δ) naturally. The considerations above show that it is the minimal diversity satisfying conditions Eq. (2). Note that if k = 2 we simply have We now make some comparisons between (E(X), δ) and the tightspan diversity of (X, δ) defined in [1]. Points in E(X) correspond to one-point extensions of the diversity (X, δ); points in the tightspan T (X) of X correspond to minimal one-point extensions of (X, δ). Thus T (X) ⊆ E(X). By Lemma 2.6 of [1], the tightspan diversity δ T equals the restriction of δ to T (X), noting that on T (X) the k different suprema we are taking the maxima over in (3) are all identical, and hence the expression simplifies.
Proof. First note that by construction we get δ(∅) = 0 and δ({f }) = 0 for any single admissible function f . If f and g are distinct members of E(X), say, f (B) > g(B) for some finite B. Then δ({f, g}) > 0.
To show monotonicity of δ, note that restricting the size of the set of elements of E(X) restricts the number of functions that can take the first position in the supremand and restricts that the corresponding A i must be the empty set. So δ can only decrease when removing elements from a set.
To show that δ satisfies the triangle inequality, let F and G be two finite sets of functions in E(X) and let h be another admissible function. Let arbitrary ǫ > 0 be given. By the definition of δ there is a collection of sets A i and B i as well an index j such that We can assume, without loss of generatlity, that the index j belongs to one of the admissible functions in F . Adding and subtracting h(∪ k B k ) gives This is true for all ǫ > 0 so the triangle inequality holds.

Extensions, supports, and E(X, ω)
Recall that a diversity (X, δ) is separable if the underlying induced metric is separable. Analogous to the metric case, the diversity (E(X), δ) need not be separable, even when (X, δ) is. To get a separable but sufficiently rich subspace of E(X) we develop the concepts of support for admissible functions of diversities.
Definition 5. Let (X, δ) be a diversity, let S ⊆ X, and let f ∈ E(S). We define the extension of f to X as The definition of f X S can be viewed as a one-point amalgamation. Amalgamation is a concept from algebra that also occurs in model theory. Two structures that share a common substructure are simultaneously embedded into a larger structure. Here the two structures are diversities. One is (X, δ), and the other is the diversity on S ∪ {z} corresponding to the function f , where z is a single point that may or may not be in S. Since S ⊆ X, the two diversities overlap (have a common substructure) on S. In Lemma 7 below, we show that f X S is an admissible function, and hence it corresponds to a diversity on X ∪{z} that extends both (X, δ) and the diversity on S ∪{z} corresponding to f . Furthermore, it is the maximal such extension. Definition 6. Let g be an admissible function on (X, δ) and S ⊆ X be nonempty. If g = f X S for some f ∈ E(S) we say that g has support S. We say that f is finitely supported if it has some finite support S.
In the following we use g ↾ S to denote the restriction of g to S.
Lemma 7. Let (X, δ) be a diversity, let S ⊆ X, and let f ∈ E(S). Then f X S is an admissible function on X, such that f X S (A) = f (A) for all finite A ⊆ S. Furthermore, it is the unique maximal such extension, in that for any admissible function g such that g ↾ S = f , we have g(A) ≤ f X S (A) for any finite set A ⊆ X. Proof. We first show that f X S is an admissible function on X by checking conditions (i) through (iv). (i) follows from the non-negativity of f and δ and setting B to be the empty set. To show (ii) we use property (ii) for f to see that the expression inside the infimum for f X S (A) satisfies where we have used condition (ii) for f and then the triangle inequality for diversities. For condition (iii), let C be an arbitrary nonempty set. Then For each such choice of {A d }, d ∈ D, let e be an element of D such that A e and C intersect. Then from the triangle inequality δ( where we have used the fact that the infimum increases because we restricted it to the case when D is a union of two sets, one of which indexes a cover of A and the other indexes a cover of B (and we've allowed some double counting of indices). Now Let f be any admissible function on (X, δ) and let S = X. Repeated use of property (iii) of admissible functions shows in equation (4), so equality holds for all A. Hence all admissible functions on (X, δ) have X as a support.
We define E(X, ω) = {f ∈ E(X) : f is finitely supported} Note that E(X, ω) is a subspace of E(X), and that κ x is finitely supported for each x ∈ X since it has support {x}. So E(X, ω) with diversity δ is still an extension of the given diversity (X, δ).
We now show that (E(X, ω), δ) is separable whenever (X, δ) is. Recall that separability of a diversity just means separability of the induced metric space. Lemma 8. Let (X, δ) be a diversity with |X| = n < ∞. Then E(X) = E(X, ω) is homeomorphic to a closed subspace of R P fin (X) .
Proof. Every function f ∈ E(X) can be naturally identified as an element of R P fin (X) . E(X) corresponds to those elements of R P fin (X) with the element satisfying the conditions of an admissible function. Since these conditions consist of a linear equality and some non-strict linear inequalities, the subset of E(X) is closed in R P fin (X) . We just need to show that the metric induced by δ is homeomorphic to the Euclidean metric.
Since δ({f, g}) = sup B finite |f (B) − g(B)| which is the ℓ ∞ norm, this gives the same topology as the Euclidean norm in R P fin (X) . Proof. Using property (iii) of admissible functions Applying the same argument with B and A reversed gives f (B) ≤ f (A) + nǫ.
To show that (E(D, ω), δ D ) is separable, note that it is the union, over all finite subsets S ⊆ D, of the extensions of (D, δ) with support S. Since each set of extensions is separable (being homeomorphic to a closed subset of a finite-dimensional Euclidean space by Lemma 8), and there are only countably many of them, (E(D, ω), δ D ) is separable.
To show that (E(D, ω), δ D ) is densely embeddable in (E(X, ω), δ X ), we define the embedding γ. For f ∈ E(D, ω) we will define γf = f : P fin (X) → R via f = f X D . From Lemma 7 we have that f is an admissible function on X, f is an extension of f , and D is a support of f .
Next we need to show that for any finite set F of admissible functions on D δ X (γF ) = δ X (F ).
First note that where we have used that D is a subset of X and that γf agrees with f on D. To show conversely that δ X (γF ) ≤ δ D (F ), we need to show that for any choice of j and finite A 1 , . . . , A k ⊆ X, we can find finite B 1 , . . . , B k so that γf j (∪ i =j B i ) is arbitrarily close to γf j (∪ i =j A i ) and γf i (B i ) is arbitrarily close to γf i (A i ) for all i = j. That such B i exist follows from the density of D in X and Lemma 9.
We have shown that the map γ : E(D, ω) → E(X, ω) is an embedding. We still need to show that it is a dense embedding. Let f ∈ E(X, ω). Suppose f has finite support S, with |S| = n and elements s 1 , . . . , s n . For any ǫ > 0, find a T ⊆ D with |T | = n elements t 1 , . . . , t n such that for any subindices i 1 , . . . , i m of 1, . . . , n we have (This is possible by Lemma 9.) Now f restricted to T is still an admissible function. We want to extend it to all of D. For any finite subset A of D, define g = (f ↾ T ) D T . By Lemma 7, g is an admissible function on (D, δ), it is an extension of f ↾ T , and it has support T . Now we let g = γg be the image of g under the embedding. We need to show that g is close to f .
The functions g, f : P fin (X) → R agree on subsets of T , but g is supported on T and f is supported on S. Let A be an arbitrary finite subset of X. Since T is finite, we have for some B ⊆ T , B = {t i 1 , . . . , t im } and {A} b∈B with can be made arbitrarily small as required.

Construction of the diversity analogue of the Urysohn metric space.
Here we define the diversity analogue of the Urysohn metric space. We show that it is the unique universal Polish diversity. We also show that it is ultrahomogenous.
Definition 11. A diversity (X, δ) has the extension property if for any finite subset F of X and any admissible function f on F , there is x ∈ X such that f (A) = δ(A ∪ {x}) for any finite A ⊆ F . The extension property is also known as the Urysohn property [4].
Definition 12. We say a diversity is Polish if its induced metric space is Polish, i.e. it is separable and complete.
Lemma 13. Let (X, δ X ) and (Y, δ Y ) be diversities where X is separable with a dense subset D X and Y is complete. Any isomorphism from D X into Y can be extended to an isomorphism from X into Y .
Proof. Let φ be an isomorphism from D X into Y . Since φ preserves the diversity it also preserves the induced metrics between the two sets and is hence a uniformly continuous map. This means we can extend it to a uniformly continuous functionφ between X and Y . To show thatφ is an isomorphism, let A ⊂ X be an arbitrary finite set, with A = {a 1 , . . . , a n }. For each k = 1, . . . , n, let a 1 k , a 2 k , a 3 k , . . . be a sequence in D X such that with a i k → a k as i → ∞. We define where we have used the uniform continuity of δ X and δ Y , by Proposition 1.
Proof. We mostly follow the proof of [4, Thm. 1.2.5]. Let {x 0 , x 1 , x 2 , . . .} be a dense set in X and let {y 0 , y 1 , y 2 , . . .} be a dense set in Y . We will define a diversity isomorphism between these dense sets and then extend it to the whole space.
At stage n > 0, suppose that φ n−1 has been defined so that {x 0 , . . . , x n−1 } ⊆ dom(φ n−1 ) and {y 0 , . . . , y n−1 } ⊆ range(φ n−1 ). If x n ∈ dom(φ n−1 ) then we let φ ′ = φ n−1 . Otherwise, let F = range(φ n−1 ) and consider the admissible function on F defined by . We extend φ n−1 to φ ′ by defining φ ′ (x n ) = y. Now if y n ∈ range(φ ′ ) then we let φ n = φ ′ and go on to the next stage. Otherwise apply the above argument to φ ′−1 and use the extension property of X to obtain an extension of φ ′ . Define φ n to be this extension.
We have thus finished the definition of φ n . Let φ be the union of all φ n we have defined. Then it has the required properties.
The completion of a diversity is defined in [8]: we take the completion of the diversity's induced metric space, and then extend the original diversity function to this larger set using continuity.
Following [7] we define the following weakened version of the extension property.
Definition 15. A diversity (X, δ) has the approximate extension property if for any finite subset F of X, any admissible function f on F , and any ǫ > 0, there is an Lemma 16. If a separable diversity has the approximate extension property, then its completion has the approximate extension property.
Proof. Let (X, δ) be a diversity that is the completion of dense subset D, where (D, δ) has the approximate extension property. Let F be a finite subset of X, f ∈ E(F ), and ǫ > 0 be given. We need to find a point y ∈ X such that |δ( Order all non-zero subsets of F , A 1 , . . . , A 2 n −1 so that if A j ⊆ A i then j ≥ i. Let ǫ 0 = ǫ/2(2 n + n). Define a bijective map γ from F to γF ⊆ D so that for all nonempty A ⊆ F , |δ(γA) − δ(A)| < ǫ 0 , which is possible by Proposition 1.
Define g : P fin (γF ) → R by g(∅) = 0 and where ǫ A i = iǫ 0 . Note that g is monotonic by construction. We claim that g ∈ E(γF ).
To show g ∈ E(γF ) we need to verify the four conditions of Lemma 3. Condition (i) (g(∅) = 0) follows by definition. To obtain condition (ii), note that for non-zero A, g(γA) = f (A) + ǫ A ≥ δ(A) + ǫ A ≥ δ(γA) − ǫ 0 + ǫ A ≥ δ(γA). For condition (iii), we first observe that for any admissible function f on F and C = ∅ we have from the triangle inequality So, given A, B, C ⊆ F , with C = ∅, where we use the fact that g is monotonic and that A ∪ B ∪ C is later than A ∪ C on the list of subsets, and so ǫ A∪B∪C + ǫ 0 ≤ ǫ A∪C . Now for condition (iv) we have So g is admissible on γF . By the approximate extension property of (D, δ), there is a point y such that |g(γA) − δ(γA ∪ {y})| ≤ ǫ/2 for all A ⊆ F . Now for any Lemma 17. Any complete diversity with the approximate extension property has the extension property.
Proof. Our proof follows that of the metric case in Theorem 3.4 of [7] and Theorem 1.2.7 of [4]. Let (X, δ) be a complete diversity with the approximate extension property. Let finite F ⊆ X be given, and let f ∈ E(F ). It suffices to show there is a sequence z 0 , z 1 , . . . in X such that for all p, |δ(A ∪ {z p }) − f (A)| ≤ 2 −p for all A ⊆ F and δ({z p , z p+1 }) ≤ 2 1−p . Since X is complete and f is continuous, the sequence will have a limit z ∈ X such that By the approximate extension property of (X, δ) we can define z 0 . To use induction, suppose we have z 0 , z 1 , . . . , z p satisfying the conditions and we need to specify z p+1 . Let . This is in an admissible function on F ∪ {z p } because it is realized by the points F ∪ {f p , f } in E(F ). So by the approximate extension property there is a z ∈ X that realizes g p with error at most 2 −(p+1) . In other words The first inequality shows that |δ(A ∪ {z}) − f (A)| ≤ 2 −(p+1) and the second inequality shows that, choosing Theorem 18. If (X, δ) is a separable diversity with the extension property then its completion also has the extension property.
Proof. Since (X, δ) has the extension property, it certainly has the approximate extension property. By Lemma 16 the completion of (X, δ) has the approximate extension property. Then by Lemma 17 the completion of (X, δ) has the extension property, being complete.
We now work towards defining a complete separable diversity with the extension property. We start with a given diversity (X, δ). We let X 0 = X, δ 0 = δ. Now, for n > 0 we inductively define (X n , δ n ) by letting X n = E(X n−1 , ω) with the diversity δ n = δ n−1 .
We define (X ω , δ ω ) to by the union of all these diversities, which is well-defined because each (X n , δ n ) is embedded in (X n+1 , δ n+1 ).
Proof. Let F be a finite subset of X ω , and let f be an admissible function on F . F must be contained in X n for some n. By construction, there is some x ∈ X n+1 such that We define the diversity (U, δ U ) to be the completion of (X ω , δ ω ) when (X, δ) is the trivial diversity of a single point. By Theorem 18, (U, δ U ) also has the extension property.
We say that a Polish diversity is universal if any separable diversity is isomorphic to a subset it.
Proof. Let (X, δ) be an arbitrary separable diversity. We construct a sequence of partial isomorphisms whose union is the desired isomorphism. Let x 0 , x 1 , x 2 , . . . be a dense sequence in X. Let y be an arbitrary point in U. Let φ 0 be defined on {x 0 } by φ 0 (x 0 ) = y. Now suppose that we have an isomorphism φ n from {x 0 , x 1 , . . . , x n } into U, with φ(x i ) = y i for i = 1, . . . , n. Define the admissible function on {y 0 , . . . , y n } for finite subset A by f (A) = δ(φ −1 n A ∪ x n+1 ). By the extension property, there is a point y n+1 in U such that δ(φ −1 n A ∪ x n+1 ) = f (A) = δ U (A ∪ y n+1 ). Define φ n+1 by extending φ n with one point with φ n+1 (x n+1 ) = y n+1 . Now take the union of all of the φ n to obtain an isomorphism between {x 0 , x 1 , x 2 , . . .} and a subset of U. By Lemma 13 this isomorphism can be extended to all of X.
A Polish diversity (X, δ) is ultrahomogeneous if given any two isomorphic finite subsets A, A ′ ⊆ X, and any isomorphism φ : A → A ′ , there is an isomorphism of (X, δ) to itself that extends φ.
Proof. This proof follows the same plan as Theorem 14. Let A, A ′ be two isomorphic subsets of U, with isomorphism φ between them. Let {x 1 , x 2 , . . .} be a dense subset of U \ A and let {y 1 , y 2 , . . .} be a dense subset of U \ A ′ . Let φ 0 = φ. Suppose we have defined φ n−1 so that it is an isomorphism and A ∪ {x 1 , . . . , x n−1 } ⊆ dom(φ n−1 ) and A ′ ∪ {y 1 , . . . , y n−1 } ⊆ range(φ n−1 ). Following the proof of Theorem 14 yields a suitable φ n . Taking the union of these φ n and applying Lemma 13 gives the desired isomorphism from U to itself that is an extension of φ.
Theorem 22. Any ultrahomogeneous, universal Polish diversity has the extension property, and is thus isomorphic to (U, δ U ).
Proof. Let (X, δ) be an ultrahomogeneous, universal Polish diversity. Let F be a finite subset of X and let f be an admissible function on F . So we can define a diversity on F ∪{z} for some z such that f (A) = δ(A∪{z}) for A ⊆ F . Since (X, δ) is universal, there is an embedding φ taking F ∪ {z} into X. Let F ′ = φ(F ). Since φ is an isomorphism from F to F ′ , there is an isomorphism φ ′ of the whole space that extends φ. Consider the point φ ′−1 (z). It satisfies the property that δ(A ∪ φ ′−1 (z)) = f (A) for all A ⊆ F .

Questions
Many questions that have been considered for the Urysohn metric space also make sense for the Urysohn diversity. For instance, it is easy to show show that (U, δ U ) is compact homogeneous, namely, any isomorphic compact subdiversities are automorphic. This follows the construction in Melleray [7,Section 4.5] for the metric case. It would be worthwhile to study the isometry group of (U, δ U ) along the lines of the results surveyed in [7,Section 4.5].