Branching geodesics of the Gromov--Hausdorff distance

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov--Hausdorff distance. We then construct branching geodesics of the Gromov--Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov--Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.


Introduction
In this paper, we denote by M the set of all isometry classes of non-empty compact metric spaces, and denote by GH the Gromov-Hausdorff distance. We refer to (M, GH) as the Gromov-Hausdorff space. We denote by Q the product space [0, 1] N of the countable copies of the unit interval. The space Q is called the Hilbert cube.
In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in (M, GH), respectively. We then show that the existence of continuum many branching geodesics passing through or avoiding sets of all spaces satisfying some of the three properties shown above, or passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces, by constructing a family of geodesics continuously parametrized by the Hilbert cube. This construction implies that for a given pair of compact metric spaces, there exists a topological embedding from the Hilbert cube into M whose image contains a pair of compact metric spaces. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.
Before precisely stating our results, we introduce basic concepts. Let N ∈ N. A metric space (X, d) is said to be N -doubling if for all x ∈ X and r ∈ (0, ∞) there exists a subset F of X satisfying that B(x, r) ⊂ y∈F B(y, r/2) and Card (F ) ≤ N , where B(x, r) is the closed ball centered at x with radius r, and the symbol "Card" stands for the cardinality. A metric space is said to be doubling if it is N -doubling for some N .
Let δ ∈ (0, ∞). A metric space (X, d) is said to be δ-uniformly disconnected if for every non-constant finite sequence {z i } N i=1 in X we have δd(z 1 , z N ) ≤ max 1≤i≤N d(z i , z i+1 ). A metric space is said to be uniformly disconnected if it is δ-uniformly disconnected for some δ ∈ (0, ∞). Note that a metric space is uniformly disconnected if and only if it is bi-Lipschitz embeddable into an ultrametric space (see [3,Proposition 15.7]).
Let c ∈ (0, 1). A metric space (X, d) is said to be c-uniformly perfect if for every x ∈ X, and for every r ∈ (0, δ d (X)), there exists y ∈ X with c · r ≤ d(x, y) ≤ r, where δ d (X) stands for the diameter. A metric space is said to be uniformly perfect if it is c-uniformly perfect for some c ∈ (0, 1).
Let (X, d) and (Y, e) be metric spaces. A homeomorphism f : X → Y is said to be quasi-symmetric if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that for all x, y, z ∈ X and for every t ∈ [0, ∞) the inequality d(x, y) ≤ td(x, z) implies the inequality e(f (x), f (y)) ≤ η(t)e(f (x), f (z)). For example, all bi-Lipschitz homeomorphisms are quasi-symmetric. Note that the doubling property, the uniform disconnectedness, and the uniform perfectness are invariant under quasi-symmetric maps. In this paper, we denote by Γ the Cantor set. David and Semmes [3] proved that if a compact metric space is doubling, uniformly disconnected, and uniformly perfect, then it is quasi-symmetrically equivalent to the Cantor set Γ equipped with the Euclidean metric ( [3,Proposition 15.11]).
Let X be a topological space. A subset S of X is said to be nowhere dense if the complement of the closure of S is dense in X. A subset of X is said to be meager if it is the union of countable nowhere dense subsets of X. A subset of X is said to be comeager if its complement is meager. A subset of X is said to be F σ (resp. G δ ) if it is the union of countably many closed subsets of X (resp. the intersection of countably many open subsets of X). A subset of X is said to be F σδ (resp. G δσ ) if it is the intersection of countably many F σ subsets of X (resp. the union of countably many G δ subsets of X).
There are some results on topological distributions in the Gromov-Hausdorff space and spaces of metrics. Rouyer [14] proved that several properties on metric spaces are generic. For example, it was proven that the set of all compact metric spaces homeomorphic to the Cantor set, the set of all compact metric spaces with zero Hausdorff dimension and lower box dimension, and the set of all compact metric spaces with infinite upper box dimension are comeager in (M, GH).
For a metrizable space X, we denote by Met(X) the space of all metrics generating the same topology of X. We consider that Met(X) is equipped with the supremum metric. In [7] and [8], the author determined the topological distributions of the doubling property, the uniform disconnectedness, the uniform perfectness, and their negations in Met(X) for a suitable space X. For example, in [7], it was proven that the set of all non-doubling metrics and the set of all non-uniformly doubling metrics are dense G δ in Met(X) for a non-discrete space X.
Let DO, UD, and UP be the sets of all doubling metric spaces, all uniformly disconnected metric spaces, and all uniformly perfect metric spaces in M, respectively. To simplify our description, the symbols P 1 , P 2 , and P 3 stand for the doubling property, the uniform disconnectedness, and the uniform perfectness, respectively. Let P be a property of metric spaces. If a metric space (X, d) satisfies the property P, then we write T P (X, d) = 1; otherwise, T P (X, d) = 0. For a triple (u 1 , u 2 , u 3 ) ∈ {0, 1} 3 , we say that a metric space ( A topological space is said to be a Cantor space if it is homeomorphic to the Cantor set Γ. The author [6] proved that for every (u, v, w) ∈ {0, 1} 3 except (1, 1, 1), the set of all quasi-symmetric equivalence classes of Cantor metric spaces of type (u, v, w) has exactly continuum many elements. In [8], the author determined the topological distribution of the set of all metrics of type (u, v, w) in Met(Γ). In this paper, we develop these results in the context of the Gromov-Hausdorff space.
Note that if there exists a curve whose length is d(x, y), then there exists a geodesic from x to y (see [1,Chapter 2]). A metric space is said to be a geodesic space if for all two points, there exists a geodesic connecting them.
Ivanov, Nikolaeva, and Tuzhilin [10] proved that (M, GH) is a geodesic space by showing the existence of the mid-point of all two points of M. Klibus [12] proved that the closed ball in the Gromov-Hausdorff space centered at the one-point metric space is a geodesic space. Chowdhury and Mémoli [2] constructed an explicit geodesic in (M, GH) using an optimal closed correspondence (see also [9]). They also showed that (M, GH) permits branching geodesics by constructing branching geodesics from the one-point metric space. Mémoli and Wan [13] showed that every Gromov-Hausdorff geodesic is realizable as a geodesic in the Hausdorff hyperspace of some metric space. For every pair of two distinct compact metric spaces, they also constructed countably many geodesics connecting that metric spaces. As a development of their results on branching geodesics, in this paper, for all pairs of compact metric spaces in M, we construct branching geodesics connecting them continuously parametrized by Q.
We say that a continuous map F : [0, 1]×Q → M is an A-branching bunch of geodesics from (X, d) to (Y, e) if the following are satisfied: (1) for every q ∈ Q, we have F (0, q) = (X, d) and F (1, q) = (Y, e); (2) for every s ∈ A and for all q, r ∈ Q we have F (s, q) = F (s, r);   Let CA denote the set of all Cantor metric space in M. Note that CA is comeger in M (see [14]). By the same method of the proof of Theorem 1.3, we obtain: Theorem 1.6. Let S be any one of X(u, v, w) for some (u, v, w) ∈ {0, 1, 2} 3 or I(D) for some dimensional function D or CA. Then for all (X, d), (Y, e) ∈ S satisfying GH((X, d), (Y, e)) > 0, there are exact continuum many geodesics from (X, d) to (Y, e) passing through S.
In this paper, a metric space is said to be infinite-dimensional if its covering dimension is infinite. Theorem 1.7. Let S be any one of X(u, v, w) for some (u, v, w) ∈ {0, 1, 2} 3 or I(D) for some dimensional function D or CA. Then for all (X, d), (Y, e) ∈ S, there exists a topological embedding from the Hilbert cube Q into S whose image contains (X, d) and (Y, e). In particular, S is infinite-dimensional.
The following theorem states that the sets I(D) and X(u, v, w) and CA are everywhere infinite-dimensional. Since all separable metrizable spaces are topologically embeddable into Q, we have: Corollary 1.9. If S is any one of X(u, v, w) for some (u, v, w) ∈ {0, 1, 2} 3 , or I(D) for some dimensional function D or CA, then every separable metrizable space X can be topologically embeddable into S. proved that all separable ultrametric spaces are isometrically embeddable into the Gromov-Hausdorff ultrametric space.
The organization of this paper is as follows: In Section 2, we prepare and explain basic concepts and statements on metric spaces. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we introduce specific versions of telescope spaces and sequentially metrized Cantor spaces introduced in [6]. Using telescope spaces, we also construct a family of compact metric spaces continuously parameterized by Q, which are not isometric to each other. In Section 5, we first prove Theorems 1.3, 1.4, and 1.5. As its applications, we next prove Theorems 1.6, 1.7, and 1.8. In Section 6, for the convenience for the readers, we exhibit a table of symbols.

Preliminaries
In this section, we prepare and explain the basic concepts and statements on metric spaces.
2.1. Generalities. In this paper, we denote by N the set of all positive integers. The symbol ∨ stands for the maximal operator of R. Let X be a set. A metric d on X is said to be an ultrametric if for all x, y, z ∈ X the metric d satisfies d(x, y) ≤ d(x, z) ∨ d(z, y). In this paper, for a metric space (X, d), and for a subset A of X, we represent the restricted metric d| A 2 as the same symbol d as the ambient metric d. For a subset A of X, we denote by δ d (A) the diameter of A, and we The following two lemmas are used to prove our results. Proof. For x ∈ [0, 1], let ξ(x) be the distance between x and A. Then the function L · ξ : Proof. We only need to consider the case of These imply the inequality.

Quasi-symmetrically invariant properties. For a metric space
By the definition of the doubling property, we obtain the following two lemmas.
Lemma 2.4. Let (X, d) be a metric space, and let A be a subset of X.
Note that the doubling property is equivalent to the finiteness of the Assouad dimension.
By definitions of ultrametric spaces, we obtain: For two metric spaces (X, d) and (Y, e), we denote by d × ∞ e the ∞ -product metric defined by Note that d × ∞ e generates the product topology of X × Y .
In this paper, we sometimes use the disjoint union i∈I X i of a non-disjoint family {X i } i∈I . Whenever we consider the disjoint union i∈I X i of a family {X i } i∈I of sets (this family is not necessarily disjoint), we identify the family {X i } i∈I with its disjoint copy unless otherwise stated. If each X i is a topological space, we consider that i∈I X i is equipped with the direct sum topology.
Next we review the basic statements of the doubling property, the uniform disconnectedness, and the uniform perfectness.
By the definition of the uniform perfectness, we obtain: We refer the readers to [5] for the details of the following: Proposition 2.11. The doubling property, the uniform disconnectedness, and the uniform perfectness are invariant under quasi-symmetric maps.
By the definitions of the doubling property, and the uniformly disconnectedness, we obtain: For a property P of metric spaces, and for a metric space (X, d) we define S P (X, d) as the set of all points in X of which no neighborhoods satisfy P (see [6,Definition 1.3]). By Proposition 2.11, we obtain: If metric spaces (X, d) and (Y, e) are quasi-symmetric equivalent to each other, then so are S P k (X, d) and This concept was introduced in the author's paper [6], and the existence of such spaces was proven in [6, Theorem 1.7].
The definition of totally exotic types and Proposition 2.12 imply: ∈ (0, ∞), and for metric spaces X and Y , a pair (f, g) with f : X → Y and g : Y → X is said to be an -approximation between (X, d) and (Y, e) if the following conditions hold: (1) for all x, y ∈ X, we have |d(x, y) − e(f (x), f (y))| < ; (2) for all x, y ∈ Y , we have |e(x, y) − d(g(x), g(y))| < ; (3) for each x ∈ X and for each y ∈ Y , we have Remark that if there exists a map f : (X, d) → (Y, e) satisfying (1) and y∈f (X) B(y, ) = Y , then there exists g : (Y, e) → (X, d) such that (f, g) is a 3 -approximation between (X, d) and (Y, e) (see [11]).
The proof of the next lemma is presented in [1] and [11].
converging to 0, and for each i ∈ N there exists an i -approximation between (X i , d i ) and (X, d).
Let (X, d) and (Y, e) be compact metric spaces. We say that a sub- where π X and π Y are projections into X and Y , respectively. We denote by R(X, Y ) (resp. CR(X, d, Y, e)) the set of all correspondences (resp. closed correspondences in The proof of the next lemma is presented in [1]. Let (X, d) and (Y, e) be compact metric spaces. We denote by CR opt (X, d, Y, e) the set of all G ∈ CR(X, d, Y, e) satisfying that dis(G) = inf R∈R(X,Y ) dis(R). An element of CR opt (X, d, Y, e) is said to be optimal. The proof of the next lemma is presented in [2] and [9]. For a set X, we define ∆ X ∈ R(X, X) by ∆ X = { (x, x) | x ∈ X }, and we call it the trivial correspondence of X.

2.4.
Amalgamation of metrics. Since the following lemma seems to be classical, we omit the proof. For n ∈ N, we put n = {1, 2, . . . , n}, and we consider that n is always equipped with the discrete topology.
Let (X, d) be a metric space and ∈ (0, ∞). A finite subset S of X is an -net if α d (S) ≥ and x∈S B(x, ) = X.
(2) For every dimensional function D, the set I(D) is dense in M.
(3) The set CA is dense in M.
The statements (2) and (3) can be proven in the same way as the statement (1). For the statement (2), we use the condition (2) in the definition of dimensional functions.

Topological distributions
In this section, we prove Theorems 1.1 and 1.2. Proof. Since M\DO is dense (see Lemma 2.20), it suffices to show that DO is F σ . For C, β ∈ (0, ∞), let S(C, β) be the set of all (X, d) such that for every finite subset of X we have We now prove that each S(C, β) is closed in M. Take a convergent sequence {(X i , d i )} i∈N in S(C, β), and let (X, d) be its limit space. Then, by Lemma 2.16, there exist a positive sequence { i } i∈N converging to 0, and sequences {f i : Take an arbitrary finite subset A of X. Take a sufficiently large i ∈ N, then g i : A → g i (A) is bijective. Thus, By letting i → ∞, we obtain Then (X, d) ∈ S(C, β), and hence S(C, β) is closed in M. By Lemma 2.3, we obtain Therefore, we conclude that DO is F σ in M.
Lemma 3.2. The set UD is F σ and meager in M.
Proof. Since M\UD is dense (see Lemma 2.20), it suffices to show that UD is F σ . For δ ∈ (0, 1), we denote by S(δ) the set of all δ-uniformly disconnected compact metric spaces. We now prove that each S(δ) is closed in M. Take a convergent sequence {(X i , d i )} i∈N in S(δ), and let (X, d) be the its limit compact metric space. Then, by Lemma 2.16, there exist a positive sequence { i } i∈N converging to 0, and sequences By letting i → ∞, we obtain δd(z 1 , z N ) ≤ max 1≤i≤N d(z i , z i+1 ). This implies that (X, d) ∈ S(δ). Since Proof. Since M \ UP is dense (see Lemma 2.20), it suffices to show that UP is F σ . For c ∈ (0, 1) and t ∈ (0, ∞), let S(c, t) be the set of all metric spaces (X, d) such that for all x ∈ X for all r ∈ (0, t) there exists y ∈ X with cr ≤ d(x, y) ≤ r. We prove that S(c, t) is closed in M. Take a convergent sequence {(X i , d i )} i∈N in S(c, t), and let (X, d) be its limit compact metric space. Then, by Lemma 2.16, there exist a positive sequence { i } i∈N converging to 0, and sequences {f i : (X i , d i ) → (X, d)} i∈N and {g i : (X, d) → (X i , d i )} i∈N such that for each i ∈ N the pair (f i , g i ) is an i -approximation. Take arbitrary z ∈ X, and for each n ∈ N put x i = g i (z). Then x i ∈ X i and d(f i (x i ), z) ≤ i . By (X i , d i ) ∈ S(c, t), there exists y i with cr ≤ d n (x i , y i ) ≤ r. Combining these inequalities and |d n ( By extracting a subsequence if necessary, we may assume that the sequence {f i (y i )} i∈N is convergent. Let w be its limit point. By letting n → ∞, by (3.1), we obtain cr ≤ d(z, w) ≤ r. This implies that (X, d) ∈ S(c, t), and hence S(c, t) is closed in M. Since we conclude that UP is F σ .
We now prove Theorems 1.1 and 1.2.
Proof of Theorem 1.2. From Theorem 1.1 and Lemma 2.20, and the fact that the intersection of an F σ set and a G δ set of a metric space is F σδ and G δσ , Theorem 1.2 follows.

Constructions of metric spaces
In this section, we prepare constructions of metrics spaces to prove Theorem 1.3.
Let X = {(X i , d i )} i∈N be a family of metric spaces satisfying the inequality δ d i (X i ) ≤ 2 −i−1 for all i ∈ N. We put and define a symmetric function d X : (T (X )) 2 → [0, ∞) by Then d X is a metric on T (X ). This construction is a specific version of the telescope space defined in [6]. The space (T (X ), d X ) is the same as the telescope space T (X , R), d (X ,R) , where R is the telescope base defined in [6, Definition 3.2]. The symbol R is a pair of the space {0} ∪ { 2 −n | n ∈ N } with the Euclidean metric and its numbering map. We review the basic properties of this construction. (2) If there exists δ ∈ (0, 1) such that each (X i , d i ) is δ-uniformly disconnected, then (T (X ), d X ) is uniformly disconnected.
is c-uniformly perfect and satisfies M · 2 −i ≤ δ d i (X i ), then (T (X ), d X ) is uniformly perfect.
We denote by 2 ω the set of all maps from Z ≥0 into {0, 1}. We define v : For each c ∈ (0, 1), we define g c : Z ≥0 ∪ {∞} → [0, ∞) by g c (n) = c n if n ∈ Z ≥0 and g c (∞) = 0. We define a metric β c on 2 ω by β c (x, y) = g c (v(x, y)). For every c ∈ (0, 1), the metric space (2 ω , β c ) is a Cantor space. This metric space is a specific version of a sequentially metrized Cantor space defined in the author's paper [6]. The following is implicitly shown in the proof of [6, Lemma 6.3].
The following proposition states the continuity of {(U (P q ), e Pq )} q∈Q .
Proof. For q ∈ Q, put (X q , d q ) = (U (P q ), e Pq ). Note that X q = X r for all q, r ∈ Q. We fix q ∈ Q, and take arbitrary ∈ (0, ∞). Since Take n ∈ N such that 2 −n ≤ min{δ, }. We denote by V the set of all r ∈ Q satisfying that for every i ∈ {1, 2, . . . , n} we obtain |q i − r i | ≤ 2 −n . Note that V is a neighborhood of q in Q. We then estimate dis(∆ Xq ). By the definition of the metric space (U (P q ), e Pq ), for every r ∈ V , the quantity |d q (x, y) − d r (x, y)| only takes values in the set This implies that dis(∆ Xq ) ≤ 2 . Thus, we conclude that for every r ∈ V , we obtain GH(G(q), G(r)) ≤ , and hence G is continuous.
A topological space is said to be perfect if it has no isolated points. For a metric space (X, d), and v ∈ [0, ∞) we denote by A(v, X, d) the set of all x ∈ X for which there exists r ∈ (0, ∞) such that for every ∈ (0, r) we have dim A (B(x, ), d) = v. Note that if (X, d) and (Y, e) are isometric to each other, then so are A(v, X, d) and A(v, Y, e) for all v ∈ [0, ∞).
Under certain assumptions on a family P, we can prove that the family {(U (P q ), e Pq )} q∈Q are not isometric to each other. Proposition 4.6. Let P = {(P i,j , p i,j )} i∈{1,2,3},j∈N be a sequence of compact metric spaces with δ p i,j (P i,j ) ≤ 2 −j−1 l(π/6). We assume that either of the following conditions holds true: (1) Each (P i,j , p i,j ) is the one-point metric space.
(2) For all i ∈ {1, 2, 3} and j ∈ N, we have: (2-a) the space (P i,j , p i,j ) is perfect; Then for all q, r ∈ Q with q = r, and for all K, L ∈ (0, ∞), the spaces (U (P q ), K · e Pq ) and (U (P r ), L · e Pr ) are not isometric to each other.
Proof. Put (X, d) = (U (P q ), K · e Pq ) and (Y, e) = (U (P r ), L · e Pr ). For the sake of contradiction, suppose that there exists an isometry I : (X, d) → (Y, e). We first assume that the condition (1) is true. By the definition, the spaces (X, d) and (Y, e) have unique accumulation points, say ω, ω , respectively. Then consists of three points, and they form an isosceles triangle. The apex angle of this isosceles triangle is equal to θ(q i ), and Then we obtain θ(q i ) = θ(r i ), and hence q = r. This is a contradiction.
Second, we assume that the condition (2) holds true. Note that both (X, d) and (Y, e) contain P i,j for all (i, j) ∈ {1, 2, 3}×N and the element represented as ∞. For each (i, j) ∈ {1, 2, 3} × N, we put W 1 (i, j) = A(dim A (P i,j , p i,j ), X, d) and W 2 (i, j) = A(dim A (P i,j , p i,j ), Y, e). By the assumption (2-b) and (2-c), and by the fact that each P i,j is open in X and Y , respectively, for all k ∈ {1, 2} we have By Definition 4.1, for all k ∈ {1, 2}, if ∞ ∈ W k (i, j), then ∞ is an isolated point of W k (i, j). Then, since each P i,j is perfect (the assumption (2-a)), for each (i, j) ∈ {1, 2, 3} × N, we can take a non-isolated point a i,j ∈ W 1 (i, j). Note that a i,j = ∞ and a i,j ∈ P i,j . Since I is an isometry and A is invariant under isometries, the point I(a i,j ) ∈ Y is a non-isolated point and I(a i,j ) ∈ W 2 (i, j). Since ∞ ∈ W 2 (i, j) or ∞ is an isolated point of W 2 (i, j), we have I(a i,j ) ∈ P i,j ⊂ Y . Put Q = {({a i,j }, d i,j )} i∈{1,2,3},j∈N , where d i,j is the trivial metric. Then (U (Q q ), e Qq ) and (U (Q r ), e Qr ) are isometric to each other. Thus by the first case, we obtain q = r. This leads to the proposition.

Geodesics
In this section, we prove Theorem 1.3 and 1.4. As its applications, we next prove Theorem 1.6, 1.7, and 1.8.
The following construction of geodesics using optimal closed correspondences is presented in [2] and [9].
The following is a useful criterion of geodesics. The proof is presented in [2, Lemma 1.3].

Proof. By Lemma 5.2, it suffices to show that
We assume that s, t ∈ [0, 1] \ A, and take an optimal correspondence R ∈ CR opt (X s , d s , X t , d t ). Put For all ((x, a), (x , a)), ((y, b), (y , b)) ∈ U , by Lemma 2.2 we obtain Since R is optimal, we obtain By the assumption, d 1 )). This implies the inequality (5.1). The remaining cases can be solved in a similar way.
Proof. Since F is continuous, by Lemma 5.2, it suffices to show the equality GH(F (s), F (t)) ≤ |s − t| · GH(X 0 , d 0 , X 1 , d 1 ) for all s, t ∈ (0, 1). Take K ∈ CR opt (R, d s , R, d t ) with (o, o) ∈ K. We define a correspondence U ∈ R(Z, Z) by The rest of the proof can be presented in the same way as the proof of Proposition 5.3. Remark that to obtain the inequality we need to use the assumption that (o, o) ∈ K.
We define a map f : By Proposition 5.3, the map f is a geodesic from (X, d) to (Y, e). Note that by the latter parts of Propositions 5.1 and 5.3, for all s, t ∈ (0, 1) the trivial correspondence ∆ W is in CR opt (f (s), f (t)).
By Proposition 4.5, we first observe that the map F is continuous. By the definition of F , the conditions (1) and (2) in Definition 1.1 are satisfied.
Since for each q ∈ Q we have δ eq (U q ) ≤ 1, and since the trivial correspondence ∆ W is in CR opt (W, E s , W, E t ), by Proposition 5.4. the map F q : [0, 1] → M is a geodesic. Thus the condition (3) in Definition 1.1 is satisfied.
We now prove the condition (4) in Definition 1.1. Before doing that, depending on (u, v, w), we define isometrically invariant operations picking out the space (U q , e q ) from F (s, q). Let (S, m) be an arbitrary compact metric space. We denote by C M (S, m) the set of all x ∈ S for which there exists r ∈ (0, ∞) such that for all ∈ (0, r) we have dim A (B(x, ), m) ≤ M . We denote by I(S, m) the set of all isolated points of (S, m). Let (u, v, w) ∈ {0, 1}, we define where CL S is the closure operator of S. Then B (u,v,w) (S, m) is an isometric invariant, i.e., if (S, m) and (S , m ) are isometric, then so are B (u,v,w) (S, m) and B (u,v,w) (S , m ). By the definitions of M , P, and Z q , and by Lemma 2.15, we see that for every (s, q) ∈ ([0, 1] \ A) × Q the space B (u,v,w) (F (s, q)) is isometric to (U q , ζ 2 (s) · e q ).
Take (s, q), (t, r) ∈ ([0, 1] \ A) × Q. We now prove that if F (s, q) and F (t, r) are isometric to each other, then (s, q) = (t, r). Under this assumption, since B (u,v,w) is invariant under isometric maps, (U q , ζ 2 (s) · e q ) and (U r , ζ 2 (t)·e r ) are isometric. Thus, by Proposition 4.6, we obtain q = r. Since F q is a geodesic, we observe that s = t. This implies the condition (4) in Definition 1.1.
Subsequently, we prove that for all (s, q) ∈ [0, 1]×Q we have F (s, q) ∈ X(u, v, w). By Lemma 2.7, we first observe that T P 1 (W, E s ) = u and T P 2 (W, E s ) = v. Then we also observe that T P 1 (W × U q , E t × ∞ (ζ 2 (s)e q )) = u and T P 2 (W × U q , E t × ∞ (ζ 2 (s)e q )) = v. Since F (s, q) is a subspace of (W ×U q , E t × ∞ (ζ 2 (s)e q ), and since F (s, q) contains (W, E s ), by Proposition 2.12, we obtain T P 1 (F (s, q)) = u and T P 2 (F (s, q)) = v. By the definitions of (U q , e q ) and F (s, q), and by Lemmas 2.9 and 2.10, we infer that T P 3 (F (s, q)) = w. Thus, we obtain F (s, q) ∈ X(u, v, w).
Therefore we conclude that Theorem 1.3 holds true.
1.3 or 1.4. We define a map G : K → X(u, v, w) by if a = 1; F (a, b/m(a)) otherwise.
Subsequently, G is continuous and injective. Since K is compact, the map G is a topological embedding. This completes the proof.

Symbol
Description M The set of all isometry classes of compact metric spaces.

GH
The Gromov-Hausdorff distance. Q The Hilbert cube. B(x, r) The closed ball centered at x with radius r.

N
The set of all positive integers. Met(X) The set of all topologically compatible metrics of a metrizable space X. P k P 1 : the doubling property, P 2 : the uniform disconnectedness, P 3 : the uniform perfectness.

DO
The set of all doubling compact metric spaces in M.

UD
The set of all uniformly disconnected compact spaces in M.

UP
The set of all uniformly perfect compact spaces in M. T P (X, d) The truth value of the statement that (X, d) satisfies a property P. Q k Q 1 = DO, Q 2 = UD, and Q 3 = UP.
The set of all (X, d) in M with D(X, d) = ∞ for a dimensional function D.

CA
The set of all compact metric spaces homeomorphic to the (middle-third) Cantor set. ∨ The maximum operator on R. ∧ The minimum operator on R.

Card(A)
The cardinality of a set A. δ d (A) The diameter of A by a metric d.

X Y
The direct sum of X and Y . i∈I X i The direct sum of a family {X i } i∈I . S P (X, d) The set of all points in (X, d) of which no neighborhoods satisfy a property P. R(X, Y ) The set of all correspondences of X and Y (see Subsection 2.3). CR(X, d, Y, e) The set of all closed correspondences in X × Y in R(X, d, Y, e). CR opt (X, d, Y, e) The set of all optimal R ∈ CR(X, d, Y, e). dis(R) dis(R) = sup (x,y),(u,v)∈R |d(x, u) − e(y, v)|. ∆ X ∆ X = { (x, x) | x ∈ X } ∈ R(X, X). n n = {1, . . . , n}. (T (X ), d X ) The telescope space for X defined in Section 4. P q P q = (P, q), where q ∈ Q and P = {(P i,j , p i,j )} i∈{1,2,3},j∈N be a sequence of compact metric spaces indexed by {1, 2, 3} × N (see Definition 4.1). (U (P q ), e Pq ) The telescope space constructed in Definition 4.1.