On the Topological and Uniform Structure of Diversities

Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note, we consider the analytical properties of diversities, in particular the generalizations of uniform continuity, uniform convergence, Cauchy sequences, and completeness to diversities.We develop conformities, a diversity analogue of uniform spaces, which abstract these concepts in the metric case. We show that much of the theory of uniform spaces admits a natural analogue in this new structure; for example, conformities can be defined either axiomatically or in terms of uniformly continuous pseudodiversities. Just as diversities can be restricted to metrics, conformities can be restricted to uniformities. We find that these two notions of restriction, which are functors in the appropriate categories, are related by a natural transformation.


Introduction
The theory of metric spaces is well-understood and forms the basis of much of, modern analysis.In 1956, Aronszajn and Panitchpakdi developed the notion of hyperconvex metric spaces [1] in order to apply the Hahn-Banach theorem in a more general setting.In fact, every metric space can be embedded isometrically in a minimal hyperconvex space, as discovered by Isbell [2] (as the "hyperconvex hull") and later by Dress [3] (as the "metric tight span").
These minimal hyperconvex spaces, or tight spans, proved to be powerful tools for the analysis of finite metric spaces.The theory of tight spans, or T-theory, is overviewed in [4].Its history, as well as applications to phylogeny, are given in [5].
In light of these applications of T-theory, Bryant and Tupper developed the theory of diversities alongside an associated tight span theory in [5].Diversities are multiway metrics mapping finite subsets of a ground space  to the nonnegative reals.The axioms were chosen based on their specific applications to phylogeny (where they had already appeared in special cases) and their ability to admit a tight span theory.This diversity tight span theory contains the metric tight span theory as a special case (using so-called diameter diversities), but it also allows new behavior which may be useful in situations such as microbial phylogeny, where the idea of a historical "phylogenetic tree" does not make sense.Several examples, along with pictures, of this phenomenon are given in [5].
A classic paper by Weil [6] developed the theory of uniform spaces, which generalize metric spaces.Uniform spaces admit notions of uniform continuity, uniform convergence, and completeness which coincide with the standard notions when metric spaces are considered as uniform spaces.This theory has been described in Bourbaki's General Topology [7] as well as Kelley's classic text [8].The metric topology can be derived purely from properties of the uniform space (via the so-called uniform topology), and in this sense uniform spaces lie "between" metric spaces and topologies.
In this note, we develop conformities, which generalize diversities in analogy to Weil's uniform space generalization of metrics.We will describe uniform continuity, uniform convergence, Cauchy sequences, and completeness for diversities, and show that these can be characterized in terms of conformities, giving an abstract framework in which to analyze the uniform structure of diversities.This is motivated by the observation that while diversities generalize metric spaces in a straightforward way (in fact they restrict to metric spaces), they can exhibit very nonsmooth behavior with respect to these spaces (cf.Theorem 1).Therefore, the existing tools for metric spaces are insufficient to get a handle on the behavior of diversities.

Preliminaries
Throughout this paper, we will denote the finite power set of a given set  by P fin () = { ⊆  : || < ∞} . (1) We begin with the Bryant-Tupper definition from [5]: a diversity is a pair (, ) where  is some set and  : If for some  ∈ P fin (), () = 0 but || > 1, we have the weaker notion of a pseudodiversity.It is shown in [5] from these axioms that if  ⊆ , then () ≤ (); that is, (pseudo)diversities are monotonic and that the restriction of a diversity to sets of size 2 forms a pseudometric (, ) = ({, }).We call this metric the induced metric of the diversity.For a metric space (, ), there are two important diversities on  having  as an induced metric as follows: when  = R  and  is the Euclidean metric; we refer to this diversity simply by diam.
(ii) The Steiner tree diversity (, ) is defined for each finite set  ⊆  as the infimum of the size of the minimum Steiner tree on .
(Recall that a Steiner tree on  is a tree whose vertex set  satisfies  ⊆  ⊆ , with each edge (, ) weighted by (, ).The size of the tree is the sum of its edge weights.) In fact, these examples are the extremes of diversity behavior relative to their induced metrics, in the sense that for any diversity (,   ) which induces a metric , we have where  is the Steiner tree diversity on (, ).This can be shown by a straightforward argument (Bryant and Tupper, upcoming).
To demonstrate the difference between the diameter and Steiner tree diversities, consider the Euclidean metric (R 3 , ).The induced metric of both the diameter and Steiner tree diversity is the Euclidean metric.For any finite set  contained in an -ball, diam() < .To contrast, in any -ball, we can find finite sets  for which () is arbitrarily large.Proof.Without loss of generality, we show that the result for -balls is about 0. For each  ∈ N, define which is a grid of points contained in the cube [0, 1/] 3 .Since there are  3 points, a minimum spanning tree connecting the members of   must have  3 − 1 edges, each of the lengths ≥ 1/ 2 , since that is the least distance between two points.Therefore, the size of the minimum spanning tree on   is at least ( 3 − 1)/ 2 , which can be taken as large as we like by taking  large enough.Since the minimal Steiner tree on   has a size of at least 0.615 times than that of the minimal spanning tree [9], we have (  ) → ∞ as  → ∞ even though diam(  ) → 0.
A similar construction for the Steiner tree diversity on R 2 gives sets of diversity (0.615 − ) for every  > 0 in every Euclidean ball.On R, the Steiner tree diversity and diameter diversity are identical.The dramatic difference between the many-point behavior of these two diversities in dimension 2 or higher demonstrates that diversities are not characterized by their induced metrics, even up to a constant.
In Section 2 and 3, we will define uniform convergence, uniform continuity, and completeness explicitly in terms of an underlying diversity, in Section 4 we will describe conformities, which abstract these properties for diversities.This is in analogy to Weil's uniformities, which abstract the same concepts for metric spaces.
With this goal in mind, we start with the following definitions: let (,   ) and (,   ) be diversities.Given From these definitions, and the axioms (D1) and (D2), it can be shown that limits are unique and every convergent sequence is Cauchy.If every Cauchy sequence is convergent, we call the diversity complete.Finally, if  :  →  is a function such that for every  > 0, there exists some  > 0 such that   () <  ⇒   (()) <  for every  ∈ P fin (), we say  is uniformly continuous.
It is not hard to see that for diameter diversities, these definitions coincide exactly with the standard ones on the induced metric.
For the second half of the paper, we will work extensively with filters, so we state the definition here: given a ground set , define a filter as a collection F of subsets of  satisfying  ∩  whenever ,  are in F, and  ∈ F whenever  ⊇  and  ∈ F. A filter base becomes a filter when all supersets of its elements are added, in which case we say the base generates the filter.
In this paper, we additionally require that ⌀ ∉ F.

Comparison with Metrics
In this section, we contrast the convergence of sequences with respect to diversities and their induced metrics.In particular, we show that although the Cauchy property for sequences is much stronger for diversities (we demonstrate a sequence which is not Cauchy with respect to a diversity, even though it is Cauchy with respect to the induced metric), completeness of a diversity is equivalent to completeness of its induced metric.This tells us that every diversity which induces a Euclidean metric (e.g., the Steiner tree diversity on R  ) is complete.
Since the set of Cauchy sequences in a diversity may be smaller than the set of Cauchy sequences of its induced metric, this may provide a simpler way to determine completeness of metric spaces.
At the end of the section, we construct the analogue of completion for diversities.

Completeness in Diversities and Metric Spaces
Theorem 2. Let (, ) be a diversity, and let  be its induced metric.If (, ) is a complete metric space, then (, ) is a complete diversity.
As mentioned, the set of Cauchy sequences in a diversity may be strictly smaller than the set of Cauchy sequences in the induced metric.For example, let (, ) be the Steiner tree diversity on R 3 , and consider the sets {  } ∈N from Theorem 1.
Order each set   somehow and define the sequence {  } by concatenating them, that is, which is Cauchy in the induced metric of (, ) (since eventually every pair of points is confined to arbitrarily small cubes [0, ] 3 ).However, it is not Cauchy in (, ), since we saw in the proof of Theorem 1 that (  ) becomes arbitrarily large as  → ∞.In other words, every tail of {  } has arbitrarily large finite sets, so {  } is not Cauchy.In light of this example, it is interesting to know that every complete diversity has a complete induced metric, which is proved with the following lemma.Lemma 3. Let (, ) be a diversity, and let  be its induced metric.Let {  } be Cauchy in (, ).Then, it has a subsequence that is Cauchy in (, ).
Theorem 4. Let (, ) be a diversity, and let  be its induced metric.If (, ) is a complete diversity, then (, ) is a complete metric space.
Proof.Let {  } be a Cauchy sequence in (, ).Then, by Lemma 3 it has a subsequence {   } that is Cauchy in (, ), which converges to some element  since the diversity is complete (it converges in both (, ) and (, )).

Completion.
In light of the equivalence between metric completeness and diversity completeness, it is perhaps not so surprising that every diversity can be completed in a canonical way.To do so, we require two more definitions from [5]: an embedding  :  1 →  2 is an injective map between diversities ( 1 ,  1 ) and ( 2 ,  2 ) such that  1 () =  2 (()) for all  ∈ P fin ( 1 ).A isomorphism is a surjective embedding.Theorem 5. Every diversity (, ) can be embedded in a complete diversity.
Proof.Let X be the set of all Cauchy sequences in .Identify any two sequences {  }, {  } which satisfy lim  → ∞ ({  ,   }) = 0 (so X is actually a set of equivalence classes).Define the function δ from P fin ( X) → R by It can then be shown that ( X, δ) is a complete diversity, and that the map   → {, , , . ..} from (, ) is an embedding.The proof is an exercise in notation.This completion is dense in the sense that every member  of X has a sequence {  } ⊆  with   →  in X (let {  } be a representative of  and define   = { 1 ,  2 , . . .,   ,   ,   , . ..}).It also satisfies a universal property analogous to that for metric completion.Theorem 6.Let (, ) be a diversity, and let ( X, δ) be its completion.Then, for any complete diversity (, ) and any uniformly continuous function  :  → , there is a unique uniformly continuous function f : X →  which extends .
Proof.Let {  } be a representative sequence of some members of X, and define f({  }) = lim  → ∞ (  ), which is defined and independent of the representative since  is uniformly continuous and  is complete.To show f is uniformly continuous, pick  > 0 and  > 0 such that (()) <  whenever () <  for all  ∈ P fin ().Then, for all  = {{ 1  }, { 2  }, . . ., {   }} ∈ P fin ( X) with δ() < /2, we have )) <  since for large enough , ({(   )}  =1 ) < 3/4.To show uniqueness of f, let ĝ be another uniformly continuous function extending  to X.For all  ∈ X, we have {  } ⊂  with   →  in X, and by uniform continuity This is a universal property in the sense that for every complete diversity X extending  and having the property, there is an isomorphism j : X → X. (Specifically, let j be the unique uniformly continuous extension of the identity map  :  → X to X .)

Conformities
In this section we introduce a generalization of diversities analogous to uniformities, which generalize metric spaces.Uniformities lie between metric spaces and topologies, in the sense that every metric space defines a uniformity, and every uniformity defines a topology (which coincides with the metric topology when the uniformity came from a metric).Uniformities characterize uniform continuity, uniform convergence, and Cauchy sequences, which are not topological concepts.
The carry-over from the metric case is natural but nontrivial, since diversities can behave differently on sets of different cardinality.Since this construction is qualitatively different from metric uniformities, it requires a different name.We asked ourselves "what would you call a uniformity that came from a diversity?",and the answer was clear, a conformity.
Throughout this section, we will give the analogous definitions and results for uniformities, using the standard treatment from Kelley [8].We begin by defining conformities and comparing them to uniformities; we show that just like uniformities, conformities have a countable base if and only if they are generated by some pseudodiversity.
We then briefly touch on the problem of completion for conformities.
Finally, we define power conformities; from a conformity defined on a set , we can construct a conformity on P fin () from which pseudodiversities can be considered uniformly continuous functions.We show that every conformity is generated by exactly the set of pseudodiversities which are uniformly continuous from its power conformity to R. This gives an equivalent definition of conformity in terms of pseudodiversities.

Conformities of Diversities.
Recall that for (, ) a metric space, {  } a sequence in , that {  } is Cauchy if and only if for each  > 0 there is some  such that every pair of points (  ,   ) with  > ,  >  has (  ,   ) < .
A similar characterization of uniform convergence of sequences of functions can be given in terms of pairs of points.From these observations arises the theory of uniformities, which is described in any standard text on analysis (cf.[7,8]).We briefly describe the theory here.For any set  define a uniformity on  as a filter U on  ×  satisfying (U1) (, ) ∈  for every  ∈ ,  ∈ U.
In particular, for any pseudometric space (, ) we can define the metric uniformity as the filter on  ×  defined by for each  > 0. We see from this example that (U1) expresses the requirement that (, ) = 0 for all  ∈ , (U2) expresses symmetry, and (U3) expresses the triangle inequality.Uniform structure can be defined entirely with respect to uniformities.For example, given sets ,  and uniformities U, V on  and , respectively, we can call a function  :  →  uniformly continuous if  −1 () ∈ U for every  ∈ V. (Here  acts on members of  componentwise.)A sequence {  } ⊂  is Cauchy if for every  ∈ U, there is some  such that pairs of elements (  ,   ) of {  } are in  whenever ,  > .It is not hard to see that for metric uniformities, these definitions coincide with the ordinary ones for metric spaces.
To abstract the uniform structure of diversities, uniformities are clearly insufficient.For one thing, since diversities map finite sets rather than pairs, we should seek a filter on P fin () rather than  × .Then symmetry is no longer required, but now monotonicity is.Finally, it is not meaningful to compose finite sets as in (U3), so we will need a diferent way to express an analogue of the triangle inequality.
Putting all this together, we define a conformity C on  as a filter on P fin () satisfying (C1) {} ∈  for every  ∈ ,  ∈ C.
(C3) For every  ∈ C, there exists some  ∈ C with  ∘  ⊆ , where in general we define Often the term conformity is also used to refer to the pair (, C).
An observation that will be necessary later (one which also holds for uniformities) is that for any  ∈ C, (∘)∘ =  ∘ ( ∘ ), so that  ∘  ∘  is defined unambiguously.To estimate the size of this, we also note that  ∘  ∘  ⊆ ( ∘ ) ∘ ( ∘ ).
As in the metric case, there is a canonical way to generate a conformity from a diversity; if  is a pseudodiversity on , we have the conformity generated by the sets for each  > 0. (This is equivalent to the one using strict inequalities, but typographically nicer.)As in the metric case, uniform structure can be defined on conformities in a way that generalizes that of diversities, let (, C) and (, D) be conformities.Then, a function  is uniformly continuous from  to  if for all  ∈ D, the set { −1 () :  ∈ } is in C. A sequence {  } on  is a Cauchy sequence if for all  ∈ C, P fin ({  } ≥ ) ⊆  for some integer .For conformities generated from diversities in the above way, these definitions coincide with those given in the previous section.
More generally, given a collection of pseudodiversities {  } ∈A , we can generate a conformity from the sets { −1  [0, ]} ∈A,>0 .We, therefore, seek a characterization of conformities in terms of the diversities which generate them.(In a later section, we will see that all conformities can be described in this way, so that we can define conformities in terms of such sets.)We begin by stating a result from Kelley [8] along with a summary of his proof.

Theorem 7. A uniformity is generated by a single pseudometric if and only if it has a countable base.
The standard proof of this theorem goes as follows: it is obvious that any uniformity generated by a pseudometric has a countable base.Conversely, if there exists a countable base for a uniformity on , there exists a countable base {  } ∈N for which the following argument holds.Define the function (, ) = 2 − , where  = sup{ : (, ) ∈   }.This generates the uniformity but does not satisfy the triangle inequality, so define where the infimum is taken over all sequences {  }  =1 with  1 =  and   = .This clearly satisfies the triangle inequality, so it just remains to be shown that  generates the uniformity.This is done by proving that (, ) ≤ (, ) ≤ 2(, ), which follows from technical constraints on {  }.
Given a conformity with a countable base {  } on a set , one might try to translate this proof directly, define a function () : P fin () → R by () = sup{ :  ∈   }, then somehow tweak  to (a) satisfy the triangle inequality and (b) generate the same conformity as .However, it appears that any direct analogue to the "infimum over all paths" strategy used in the metric case (there are several) cannot satisfy both (a) and (b) simultaneously.
Nonetheless, the result is true, which is the content of the next theorem.Lemma 8. Let (, C) have a countable base.Then, it has a countable base {  } satisfying  0 = P fin (),   ∘  ∘  ⊆  −1 for  > 0.
Proof.Let {  } be a countable base for C. Define  0 = P fin (),   =   ∩  −1 .Then, {  } is a nested countable base.Finally, choose {  } as   =    , where   are chosen inductively as  0 = 0, then ( Theorem 9. Let (, C) be a conformity.There exists, a pseudodiversity  which generates C if and only if C has a countable base.
Proof.If  exists, the sets { 1/ } ∈N are our base.
Trivially,  ≤   .For the other inequality, choose  ∈ P fin ().Our strategy is to induct on the greatest integer  such that () < 2 − .

Induced Uniformities and Completeness.
Given a conformity C, we define its induced uniformity as the uniformity generated by the sets Next, we give some standard definitions.For a uniform space (, U), the uniform topology of U on  is the smallest topology containing the sets  (, ) = { : (, ) ∈ } , for all  ∈ ,  ∈ U. Notice that if U is generated by a pseudometric, this coincides with the pseudometric topology.
With the same space (, U), we call a filter F on  Cauchy if for every  ∈ U, there is some  ∈ F with  ×  ⊆ .We say that F converges to some  ∈  if every neighborhood of  (in the uniform topology) is in F. We then call a uniformity complete if every Cauchy filter converges.It can be shown that a metric space is complete if and only if its generated uniformity is, and that every uniformity can be embedded minimally (i.e., satisfying a universal property with respect to uniformly continuous maps) in a complete uniformity [7,8].
The analogous definitions for conformities are as follows.
Let  be a filter on .If for all  ∈ C, there exists  ∈  with P fin () ⊆ , then  is a Cauchy filter.If  ∈  and for all  ∈ C there exist  ∈  with P fin () ⊆ { :  ∪ {} ∈ }, then  converges to .Finally, if every Cauchy filter converges to some point in , we say C is complete.Theorem 12.A pseudodiversity (, ) is complete if and only if its conformity C is.
Conversely, suppose that every Cauchy filter converges in C, and let {  } be a Cauchy sequence in (, ).Choose the sets   = {  } ∞  .These sets generate a Cauchy filter with some limit .It is clear that   → .
For any conformity (, C) generated by a diversity, the conformity is complete if and only if the diversity is.The diversity is complete if and only if its induced metric is, which in turn is complete if and only if its uniformity is [7,8]; thus completeness of the conformity is equivalent to completeness of its induced uniformity.In fact, this is true in general, as the next theorem shows.
Theorem 13.Let (, C) be a conformity with complete induced uniformity U.Then, C is complete.
Proof.Suppose that U is complete, and let F be a Cauchy filter with respect to C.Then, F is also Cauchy with respect to U, since for all  ∈ C, we have {{, } : ,  ∈ } ⊆ P fin () ⊆  for some  ∈ F; then  ×  ⊆   .Thus, F converges in U to some element , and we claim that it also converges to  in C. To this end, fix  ∈ C. Choose  ∈ C so that  ∘  ⊆  and  ∈ F so that (a)  ∈  whenever (, ) ∈   and (b) P fin () ⊆ .Then, for all  ∈ P fin (),  ∪ {} ∈ .(If  = ⌀,  ∪ {} ∈  trivially.Otherwise, pick  ∈ , and we will have  ∈  and {, } ∈ , so that  ∪ {, } =  ∪ {} ∈ .) We end this section with two open questions as follows.
(1) Does the converse to Theorem 13 holds; that is, if a conformity (, C) is complete, must its induced uniformity be?
(2) We saw in Section 3.2 that for any diversity (, ), it is possible to embed  in a complete diversity which was universal, meaning that any uniformly continuous map from  to a complete diversity is factored through the embedding.It is shown in [8] that every uniformity can be embedded in a complete uniformity.This embedding is also universal.Is there a notion of universal completion for conformities?
In this section, we will show that every conformity is generated by the collection of pseudodiversities which are uniformly continuous with respect to it, in an appropriate sense.In the case of uniformities, this is done by constructing a so-called product uniformity; given a uniformity on a set , the product uniformity is constructed on ×.Then, a given pseudometric  may or may not be uniformly continuous from the product uniformity to the Euclidean uniformity on R. It can be proven [7,8] that a uniformity U is exactly the uniformity generated by all pseudometrics which are uniformly continuous from its product uniformity.

Theorem 1 .
The Steiner tree diversity function  on R 3 is unbounded on every open set of the Euclidean topology.