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Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by C\'{u}th, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of an orbit equivalence relation. It is well known that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. We prove a version of this result for pseudometrics, showing that the Gromov-Hausdorff distance of Polish metric spaces is bireducible with a universal element in a certain class of orbit pseudometrics.


Introduction and main results
Let E, F be equivalence relations on Polish spaces X, Y (see Section 2 for more details on the terminology). We say that E is Borel reducible to F if there exists a Borel mapping f : X → Y (so-called reduction) such that for all x, x ′ ∈ X. In some sense, this means that E is at most as complex as F , or that the problem of deciding whether two objects are E-equivalent can be reduced to the problem of deciding whether two objects are F -equivalent.
The study of reducibility between definable equivalence relations, often called invariant descriptive set theory, is an active area of current research, see e.g. [6]. Let us emphasize here the result of Clemens, Gao and Kechris [3,7] which states that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. Here, an orbit equivalence relation is the relation defined by xE X G y ⇔ ∃g ∈ G : g · x = y for some Borel action of a Polish group G on a Polish space X, and an orbit equivalence relation E X G is called universal if any other orbit equivalence relation E Y H is Borel reducible to E X G . It is worth noting that the isometry of Polish metric spaces is Borel bireducible also with the linear isometry of separable Banach spaces [12], with the isomorphism of separable C*-algebras [13] and with the homeomorphism of compact metric spaces [14].
Recently, the notion of Borel reducibility has been investigated by Cúth, Doucha and Kurka in the setting of pseudometrics on Polish spaces, see [4,5]. This is a more general setting, since an equivalence relation E can be viewed as the pseudometric ̺ E defined by ̺ E (x, y) = 0 if xEy and ̺ E (x, y) = 1 otherwise. Moreover, in [4], it is shown that orbit equivalence relations have a natural generalization in the setting of pseudometrics, so-called orbit pseudometrics.
Let ̺ X and ̺ Y be pseudometrics on Polish spaces X and Y . We say that ̺ X is Borel-uniformly continuous (Borel-u.c.) reducible to ̺ Y if there exists a Borel mapping f : X → Y such that, for every ε > 0, there are δ X > 0 and δ Y > 0 This is a quantitative version of the Borel reducibility of the relation ̺ X (x, x ′ ) = 0 to the relation ̺ Y (y, y ′ ) = 0. Let us accentuate that pseudometrics in this work are allowed to attain ∞ as a value.
The central object studied in [4,5] is the Gromov-Hausdorff distance of Polish metric spaces, which, in some sense, measures how far are two spaces from being isometric (see Section 2 for the precise definition, as well as for the definitions of the Urysohn space U and of the coding F (U) of Polish metric spaces, enabling us to see the Gromov-Hausdorff distance as a pseudometric ̺ GH on a Polish space). It is shown that the Gromov-Hausdorff distance is bireducible with several other distances, for instance the Lipschitz distance of Polish metric spaces or the Kadets and Banach-Mazur distances of separable Banach spaces. Also, it is bireducible with a simple looking orbit pseudometric, we recall this in Theorem 2.3.
We now formulate the main result of this work. This is a continuous version of the result of Clemens, Gao and Kechris, in which the Gromov-Hausdorff distance takes over the role of the isometry relation.
Theorem 1.1. Let G be a Polish group acting continuously on a Polish space X. Let d be a lower semicontinuous pseudometric on X such that d(x, y) = d(gx, gy) for any x, y ∈ X and g ∈ G. Moreover, let there be a system Φ of continuous pseudometrics on X such that d = sup ϕ∈Φ ϕ.
Then ̺ G,d is Borel-u.c. reducible to the Gromov-Hausdorff distance ̺ GH .
It is not clear if the theorem holds without the assumption of the existence of a system Φ of continuous pseudometrics (this is extensively discussed in Section 4). Actually, we prove a version of the theorem with a more general system Φ and a stronger conclusion, see Theorem 3.1. The constructed reduction f : Thus, in the terminology of [4], the reduction is Borel-Lipschitz on small distances.
Let us recall a remarkable problem posed by Ben Yaacov, Doucha, Nies and Tsankov [2]. Let E GH be the relation given by Then any orbit equivalence is Borel reducible to E GH , but it is not known if E GH is Borel reducible to an orbit equivalence, see [2,Question 8.6]. However, E GH possesses some properties of orbit equivalences, as its equivalence classes are Borel, see [2,Corollary 8.3] (or [4, Corollary 52]), and the relation is not Borel reducible to it, see [4,Theorem 24]. Therefore, Theorem 1.1 has the following consequence.
It turns out that our methods moreover provide a positive answer to Question 41 from [4]. Let us recall that Gao and Kechris [7] proved that the relation induced by the canonical action of the isometry group Iso(U) on F (U) is a universal orbit equivalence relation (and so it is Borel bireducible with the isometry relation of Polish metric spaces). Considering the Hausdorff distance ̺ H , we can obtain the following analogue for pseudometrics.
This theorem is also a consequence of Theorem 3.1, which provides a reduction of an orbit pseudometric to both ̺ GH and ̺ Iso(U),̺H . Similarly as in Theorem 1.1, it is possible to construct both reductions in Theorem 1.3 to be Borel-Lipschitz on small distances.

Preliminaries
Let ̺ H (A, B) denote the Hausdorff distance of two non-empty subsets A and B of a metric space. The Gromov-Hausdorff distance of non-empty metric spaces M and N is defined by where the symbol ֒→ denotes an isometric embedding. Actually, in this work, both the Hausdorff distance ̺ H and the Gromov-Hausdorff distance ̺ GH are considered mainly between elements of the space F (U) \ {∅} defined below, and both distances will be regarded as pseudometrics on this space. A binary relation R ⊆ X × Y is called a correspondence between X and Y if ∀x ∈ X∃y ∈ Y : xRy and ∀y ∈ Y ∃x ∈ X : xRy. Let (M, δ M ) and (N, δ N ) be metric spaces and let r > 0. It is well known and easy to show that if ̺ GH (M, N ) < r, then there exists a correspondence R between M and N such that A Polish space (topology) means a separable completely metrizable topological space (topology), and a Polish metric space means a separable complete metric space. A Polish group is a topological group whose topology is Polish. By an action of a group G on a set X we mean a mapping (g, x) ∈ G × X → g · x ∈ X satisfying 1 G · x = x and (gh) · x = g · (h · x) for all g, h ∈ G and x ∈ X.
The Urysohn space is defined as the (up to isometry) only Polish metric space U with the property that for any finite metric space A and any isometric embedding i : B → U, where B ⊆ A, there exists an isometric embedding i : A → U extending i. It is well known that the Urysohn space contains an isometric copy of every Polish metric space, in fact, the following result holds.
Theorem 2.1 (Katětov [9]). Let X be a Polish metric space. Then there exists an isometric embedding i : X → U such that any surjective isometry on i(X) can be extended to a surjective isometry on U.
For a Polish metric space (X, δ X ), we define F (X) = F ⊆ X : F is closed , and we equip F (X)\{∅} with the Wijsman topology, defined as the coarsest topology for which the function F → δ X (x, F ), is continuous for each x ∈ X. If we moreover add ∅ as an isolated point, then F (X) is a Polish space whose Borel σ-algebra is the Effros Borel structure of X, defined as the σ-algebra generated by the sets {F ∈ F (X) : F ∩ U = ∅}, U ⊆ X open, see e.g. [1]. Let us note that if X is universal (in the sense that it contains an isometric copy of every Polish metric space), then F (X) is a coding of all Polish metric spaces up to isometry. For this purpose, we will employ the Urysohn space and the coding F (U). Although this is not the same coding as the coding considered in the preceding works [4] and [5], it makes no important difference.
For a Polish metric space X, we denote by Iso(X) the group of all surjective isometries on X with the topology of pointwise convergence. This is a Polish group, and by its canonical action on F (X) we mean the action I · F = I(F ). Let us prove a simple lemma. Proof. Assuming I n → I and F n → F , we show that I n (F n ) → I(F ). Given x ∈ X, we need to check that δ X (x, I n (F n )) → δ X (x, I(F )). We have Concerning the moreover part, it is sufficient to notice that Let us consider the action of the group S ∞ of all permutations of N on the space Let us recall one more result which will play an important role in our construction.
Theorem 2.4 (Melleray [11]). Let X be a Polish metric space of diameter at most 1 and let G be a closed subgroup of Iso(X). Then there exists an extension Y of X such that • Y is a Polish metric space, • any member of G can be extended in a unique way to a surjective isometry on Y , • any surjective isometry on Y is an extension of a member of G.
Since this theorem is not stated explicitly in [11], it is necessary to give an explanation. Theorem 2.4 can be proved in the same way as [11, Theorem 1.1] with the following two differences: • The part before Claim is to be ignored.
• We want to preserve the distance, so d should not be replaced by d 1+d after the proof of [11,Lemma 3.2]. The purpose of this replacement is to fulfill diam(Z) ≤ 1. However, it holds for the original distance that diam(Z) ≤ 3, which is sufficient for the final step of the construction. Therefore, d can be preserved.

The Construction
The aim of this section is to prove the following theorem from which the results introduced above follow.
Theorem 3.1. Let G be a Polish group acting continuously on a Polish space X. Let d be a lower semicontinuous pseudometric on X with d ≤ 1 such that d(x, y) = d(gx, gy) for any x, y ∈ X and g ∈ G. Moreover, let there be a system Φ of lower semicontinuous pseudometrics on X such that d = sup ϕ∈Φ ϕ and every ϕ ∈ Φ has the property that the mapping for all p, q ∈ X.
Before proving this theorem, let us show first that Theorems 1.1 and 1.3 are its consequences. It is sufficient to prove Theorem 1.1 in the case d ≤ 1, as one can consider min{d, 1} instead of d. So, we just need to show that any continuous pseudometric ϕ on X satisfies the requirement from Theorem 3.

For an open
Let us assume that p ∈ X is such that H ϕ (p) intersects U , and let us pick (x, u) ∈ H ϕ (p) ∩ U . If u < 1, consider v > u such that (x, v) ∈ U , and if u = 1, consider v = 1. As ϕ is continuous, for every q from a neighborhood of p, we have ϕ(q, x) ≤ v, and so (x, v) ∈ H ϕ (q) ∩ U .
Concerning Theorem 1.3, the reducibility of ̺ GH to ̺ Iso(U),̺H follows from Theorem 3.1 and Theorem 2.3, and the reducibility of ̺ Iso(U),̺H to ̺ GH follows from Theorem 1.1 and Lemma 2.2.
Let us now turn to the proof of Theorem 3.1. We notice first that there is a y)}, and due to the Lindelöf property, countably many of them form a covering as well. We choose a sequence ϕ 1 , ϕ 2 , . . . in Φ ′ such that every element of Φ ′ appears infinitely many times.
Let γ be a compatible right-invariant metric on G with γ ≤ 1 (let us note that the space (G, γ) needs not to be complete). Let δ X be a compatible complete metric on X with δ X ≤ 1. Let us consider the maximum metric on G × X, i.e., the metric δ G×X ((g, x), (h, y)) = max{γ(g, h), δ X (x, y)}. It is easy to check that the mapping ) is an isometry on G × X for every h ∈ G. We obtain the following claim based on Melleray's result.
• Z is a Polish metric space of diameter at most 2, • for any h ∈ G, the isometry I h : (g, x) → (gh, x) can be extended in a unique way from G × X to a surjective isometry on Z, • any surjective isometry on Z is an extension of I h for some h ∈ G.
Proof. Let G denote the completion of (G, γ). For every h ∈ G, let I * h be the unique surjective isometry on G × X extending I h . We show that the set {I * h : h ∈ G} forms a closed subgroup of Iso(G×X). This is clearly a subgroup, as I * h ′ •I * h = I * hh ′ , and by the fact that any Polish subgroup of a Polish group is closed (see e.g. [10, Exercise 9.6]), it is sufficient to show that h → I * h is a homeomorphism from G into Iso(G × X).
First, let h n → h. Then I * hn (g, x) = (gh n , x) → (gh, x) = I * h (g, x) for every (g, x) ∈ G × X. So, I * hn converges pointwise to I * h on a dense subset of G × X. As the mappings I * hn and I * h are isometries, they are equicontinuous, and the pointwise convergence on the whole G × X follows. Conversely, let I * So, {I * h : h ∈ G} is a closed subgroup of Iso(G × X) indeed, and we obtain from Theorem 2.4 an extension Z of G × X satisfying all desired properties with the only possible exception of the diameter requirement. However, such property can be easily arranged by changing the metric appropriately (we can take an increasing concave function ξ : We define a compatible complete metric m on Y in three steps. Concerning the verification of the triangle inequality, some details will be left to the reader. The first step, perhaps the least obvious one, uses a similar idea as the proofs of [5, Theorem 39] and [5,Theorem 41]. The choice of the distances between points in Z × N will play a crucial role in the proof of Claim 3.7. (i) On the subset Z × N of Y , we put m((z 1 , k 1 ), (z 2 , k 2 )) = 100 · |2 k1 − 2 k2 | + 2 min{k1,k2} δ Z (z 1 , z 2 ).
To check the triangle inequality, one can for instance proceed similarly as in the previous step, considering a suitable subset of (Z × N) × k∈N ({−10 · 2 k } ∪ [0, 1]) with the sum metric.
Once we have defined the space Y and its metric m, we can consider for every p ∈ X the subspace For every p ∈ X, let Y p be the closure of W p in Y . We now provide a series of claims concerning the spaces Y p .
Given an open U ⊆ Y , we need to check that the set {p ∈ X : Y p ∩ U = ∅} is Borel. Let us show first that for each k ∈ N, the set By the assumption on the pseudometrics in Φ, the set of all p ∈ X with this property is Borel.
Claim 3.4. For all p ∈ X and h ∈ G, there is I ∈ Iso(Y ) that maps Y p onto Y hp .
Proof. If we denote by I * * h the unique isometry on Z extending I h , then the mapping (z, k) → (I * * h (z), k), (z, u, k) → (I * * h (z), u, k), (in part. (g, x, u, k) → (gh, x, u, k)), is an isometry on Y which maps W hp onto W p . It follows that it maps Y hp onto Y p as well.
Claim 3.5. For all p, q ∈ X, Proof. Let us note that the Hausdorff distance between Y p and Y q is at most d(p, q), which follows from the fact that |ϕ k (gp, x) − ϕ k (gq, x)| ≤ ϕ k (gp, gq) ≤ d(gp, gq) = d(p, q) for every g ∈ G, x ∈ X and k ∈ N. So, using Claim 3.4, we obtain for , q), and it is sufficient to consider the infimum over h ∈ G. Claim 3.6. Let p ∈ X, w ∈ W p , and let A(w, p) = {m(w ′ , w) : w ′ ∈ W p }. Let further B ⊆ [0, ∞) be such that ̺ H (B, A(w, p)) ≤ 1 and β l be defined by β l = inf(B ∩ (80 · 2 l , ∞)) for l ∈ N.
Concerning the consequence part, we can choose the same set B for both points w, w ′ (the choice B = A(w, p) works, as well as the choice B = A(w ′ , q)). It is then sufficient to note that B determines to which one of the sets Z × {k} or Z × [0, 1] × {k} the points w, w ′ belong, as the intervals [100 · 2 k − 1, 100 · 2 k + 1] together with the intervals [90 · 2 k − 2, 90 · 2 k + 1] are pairwise disjoint.
Proof. Given r > ̺ GH (Y p , Y q ), we need to show that 2r ≥ ̺ G,d (p, q). Since r > ̺ GH (Y p , Y q ) = ̺ GH (W p , W q ), there is a correspondence R between W p and W q such that aRa ′ & bRb ′ ⇒ |m(a, b) − m(a ′ , b ′ )| ≤ 2r. As ̺ G,d (p, q) ≤ 1, we can assume that 2r < 1. In such a case, due to Claim 3.6, elements of Z × {k} may correspond only to elements of Z × {k} itself, and the same holds for Z × [0, 1] × {k}. Consequently, the relations are correspondences on Z.
So, we have shown that I is a surjective isometry on Z, and it follows that I is the extension of I h for some h ∈ G. Now, let k ∈ N be arbitrary. If we choose z such that (1 G , p)R k z and g, x, u such that (1 G , p, 0, k)R(g, x, u, k), then m((1 G , p, k), (1 G , p, 0, k)) − m((z, k), (g, x, u, k)) ≤ 2r.
As (g, x, u, k) ∈ W q , we have ϕ k (gq, x) ≤ u ≤ 1, and thus So, we have seen that for every k ∈ N, there are g k ∈ G and x k ∈ X such that We have g k → h, x k → p, and for every ϕ ∈ Φ ′ , we get ϕ(hq, p) ≤ 2r, as it is lower semicontinuous and ϕ = ϕ k infinitely many times. It follows that ̺ G,d (q, p) ≤ d(hq, p) ≤ 2r.
To finish the proof of Theorem 3.1, we apply Theorem 2.1. We may suppose that Y is actually a subspace of U such that any surjective isometry on Y can be extended to a surjective isometry on U. It is easy to check that for all p, q ∈ X. Together with Claims 3.5, 3.7 and 3.3, this shows that the mapping p → Y p works, which completes the proof of Theorem 3.1.
Let us remark here that for all p ∈ X and h ∈ G, there is I ∈ Iso(U) that maps Y p onto Y hp (this follows from Claim 3.4). Hence, our reduction has the property that Iso(U) Y q ⇒ Y p and Y q are isometric. Moreover, if d is the discrete metric, then this holds with equivalences. In this way, we obtain the result of Clemens, Gao and Kechris that E X G is Borel reducible to the relation of isometry on F (U), as well as to the relation E F (U) Iso(U) , see [3,7].

Comments and questions
An immediate question concerning Theorem 1.1 remains open. We will show in the next remark that the existence of such a system is not guaranteed for a lower semicontinuous pseudometric d. However, we do not know if the more general system from Theorem 3.1 needs to exist. We just know that the mapping p → H ϕ (p) needs not to be Borel for a lower semicontinuous pseudometric ϕ, see Remark 4.3. Later, we will show that Question 4.1 has a positive answer in the case of the trivial group action, see Proposition 4.4.
Let us recall that an equivalence relation E on a Polish space X is called smooth if there exists a Borel reduction of E into the relation of equality on a Polish space Y . It is known that every closed (actually every G δ ) equivalence relation is smooth, see [8,Corollary 1.2]. It turns out that the question if a system Φ from Theorem 1.1 exists for any lower semicontinuous pseudometric is connected with the question if the reduction witnessing smoothness of a general closed equivalence relation can be found not only Borel, but even continuous. We show in the next remark that the question has a negative answer. Although this is not a surprising finding, we were not able to find a counterexample in the literature. : k ∈ N} .
The distance of two points from different copies of A or B can be defined as 10, say. Finally, we consider the equivalence relation E with the equivalence classes 2n }, n, m ∈ N. Let us check that E is closed. Suppose that a i → a and b i → b in X and that a i Eb i for every i ∈ N. We need to check that aEb. We may suppose that each a i (each b i ) belongs to the same copy of A or B as a (as b). In the case that a and b belong to the same copy of A or B, then a i = b i for i ∈ N, and so a = b. In the opposite case, there are four possibilities, but the argument is the same in each of them. If, say, there is m such for every i, we have a i = x  for some n(i), then there is n such that n(i) = n for all but finitely many i's (as the distance of x  (1) works. Let f : X → Y be a continuous mapping into a Polish space Y such that xEy ⇒ f (x) = f (y). We show first that f (p (m) ) → f (x (0) ). Let U be a neighborhood of f (x (0) ). There is m 0 such that f (x) ∈ U whenever the distance of x to x (0) is less than 1/m 0 . For all m > m 0 and n ∈ N, we have f (p (m) n,m ) ∈ U . It follows that f (p (m) ) ∈ U for all m > m 0 . As U was an arbitrary neighborhood of f (x (0) ), we finally arrive at f (p (m) ) → f (x (0) ). By the same argument, we can obtain f (p (m) ) → f (x (1) ). Therefore, f (x (0) ) = f (x (1) ).
Remark 4.3. Concerning the assumptions of Theorem 3.1, we can ask if the mapping p → H ϕ (p) is Borel for a general lower semicontinuous pseudometric ϕ. If we restrict our attention on pseudometrics attaining only values 0 and 1, we actually ask if the mapping x → [x] E from X to F (X) is Borel for a general closed equivalence relation E on a Polish space X. This question has a negative answer. We can consider any closed equivalence without a Borel selector, or we can consider the following example due to Noé de Rancourt.
Let A be an analytic non-Borel set in a Polish space Y . Then A is the projection of a closed subset B of Y × N N . Let us add an isolated point p to N N and consider the space X = B ∪(Y × {p}) and the equivalence (y 1 , ν 1 )E(y 2 , ν 2 ) ⇔ y 1 = y 2 . Then B is an open subset of X but the set {y ∈ Y : [(y, p)] E ∩ B = ∅} = A is not Borel.
In the next proposition, we give a positive answer to Question 4.1 in the case of a trivial group action. The proposition can be seen as a continuous version of the above mentioned result from [8] that every closed equivalence is smooth. Its proof is inspired by the proof of [8, Lemma 5.1] and a proof of Urysohn's lemma. In other words, there is a Borel mapping f : X → R N such that f (p) − f (q) ∞ = d(p, q) for all p, q ∈ X.
Proof. For A ⊆ X and r > 0, let us denote (A) r = {x ∈ X : d(x, A) < r}, where d(x, A) means the distance of x to A in the pseudometric space (X, d). It holds that A is analytic ⇒ (A) r is analytic, since (A) r is a projection of the intersection of X × A with the F σ set {(x, y) ∈ X × X : d(x, y) < r}.
Let us pick η > 0 and p, q ∈ X with d(p, q) > η. We can choose open neighborhoods U ∋ p and V ∋ q such that d(x, y) > η for x ∈ U and y ∈ V . By a recursive procedure, we will find for every rational number u ∈ [0, 1] a Borel set B u in the way that means the distance of these two sets in the pseudometric space (X, d). Note first that d(B 0 , X \ B 1 ) = d(U, V ) ≥ η = η · (1 − 0). Let u 1 = 0, u 2 = 1, and let the rational numbers in (0, 1) be enumerated by u 3 , u 4 , . . . . Suppose that n ≥ 2 and that B u1 , . . . , B un are already chosen. Let us consider As C and D are disjoint analytic sets, using the Lusin separation principle (see e.g. [10,Theorem 14.7]), we can find a Borel set B un+1 which separates C from D, that is, C ⊆ B un+1 ⊆ X \ D. It is easy to check that the procedure works.
Once the sets B u are found, we define It is easy to show that s is a Borel function and that s(y) − s(x) = η − 0 = η for x ∈ U and y ∈ V . Let us check that |s(x) − s(y)| ≤ d(x, y) for all x, y ∈ X. We can suppose that s(x) > s(y). Let u, v be rational with s(y)/η < u < v < s(x)/η. Then y ∈ B u and x / ∈ B v , and so d(x, y) ≥ d(B u , X \ B v ) ≥ η · (v − u). Since u can be arbitrarily close to s(y)/η and v can be arbitrarily close to s(x)/η, we obtain d(x, y) ≥ s(x) − s(y). Now, let η > 0 be fixed. The construction above provides a covering of the set {(p, q) ∈ X × X : d(p, q) > η} by open sets of the form U × V , for each of which there is a Borel function s : X → R such that |s(x) − s(y)| ≤ d(x, y) for all x, y ∈ X and s(y) − s(x) = η for x ∈ U and y ∈ V . Due to the Lindelöf property, there is a countable system S η of such functions satisfying d(p, q) > η ⇒ sup s∈Sη |s(p) − s(q)| ≥ η. Finally, the system S = {S η : η > 0, η ∈ Q} has the desired property d(p, q) = sup s∈S |s(p) − s(q)| for all p, q ∈ X.
Concerning the second part of the proposition, for functions s n as in the first part, the mapping f (x) = (s 1 (x), s 2 (x), . . . ) works. Concerning the consequence part, we note that the pseudometric (a, b) → a − b ∞ on R N is Borel-u.c. reducible to ̺ GH by Theorem 1.1.