FRACTAL DIMENSIONS IN THE GROMOV–HAUSDORFF SPACE

. In this paper, we ﬁrst show that for all four non-negative real numbers, there exists a Cantor ultrametric space whose Hausdorﬀ dimension, packing dimension, upper box dimen-sion, and Assouad dimension are equal to given four numbers, respectively. Next, by constructing topological embeddings of an arbitrary compact metrizable space into the Gromov–Hausdorﬀ space using a direct sum of metrics spaces, we prove that the set of all compact metric spaces possessing prescribed topological dimension, and four dimensions explained above, and the set of all compact ultrametric spaces are path-connected and have inﬁ-nite topological dimension. This observation on ultrametrics provides another proof of Qiu’s theorem stating that the ratio of the Archimedean and non-Archimedean Gromov–Hausdorﬀ distances is unbounded.


Introduction
In the present paper, we mainly deal with the topological dimension dim T X, the Hausdorff dimension dim H (X, d), the packing dimension dim P (X, d), the upper box dimension dim B (X, d), and the Assouad dimension dim A (X, d) of a metric space (X, d).The definitions of these dimensions will be presented in Section 2. The topological dimension takes values in Z ≥−1 ∪ {∞}, and the other four dimensions take values in [0, ∞].For every bounded metric space (X, d), we have the following basic inequalities (see Theorem 2.8): 1.1.Fractal dimensions.A topological space is said to be a Cantor space if it is homeomorphic to the Cantor set.A metric d : X 2 → [0, ∞) on a set X is said to be a non-Archimedean metric or ultrametric if for all x, y, z ∈ X the strong triangle inequality: d(x, y) ≤ d(x, z) ∨ d(z, y) is satisfied, where ∨ is the maximal operator on R.
In [12], the author proved that for all a, b ∈ [0, ∞] with a ≤ b, there exists a Cantor metric space (X, d) with dim H (X, d) = a and dim A (X, d) = b.As a development of this result, we solve the problems of prescribed dimensions for the five dimensions explained above.
Theorem 1.1.For every (a 1 , a 2 , a 3 , a 4 ) ∈ L, there exists a Cantor ultrametric space (X, d) such that By Theorem 1.1, we obtain the following (see also Theorem 2.8): Theorem 1.2.For every (l, a 1 , a 2 , a 3 , a 4 ) ∈ R, there exists a compact metric space (X, d) such that 1.2.Topological embeddings of the Hilbert cube.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 (the definition will be presented in Section 2).The space (M, GH) is called the Gromov-Hausdorff space.By abuse of notation, we represent an element of M as a pair (X, d) of a set X and a metric d rather than its isometry class.
For (l, a 1 , a 2 , a 3 , a 4 ) ∈ R, we denote by D(l, a 1 , a 2 , a 3 , a 4 ) the set of all compact metric spaces in M satisfying dim T X = l, dim H (X, d) = a 1 , dim P (X, d) = a 2 , dim B (X, d) = a 3 , dim A (X, d) = a 4 .
We denote by U the set of all compact ultrametric spaces in M.
We also define Q = i∈Z ≥0 [0,1].The space Q ( or a space homeomorphic to Q) is called the Hilbert cube.
In [13], the author defined X(u, v, w) for (u, v, w) ∈ {0, 1, 2} 3 as sets of all compact metric spaces satisfying or not satisfying the doubling property, the uniform disconnectedness, and the uniform perfectness, which are properties appearing in the David-Semmes theorem [5,Proposition 15.11].The author [13] proved that for all compact metric spaces (X, d) and (Y, e) in X(u, v, w) with GH((X, d), (Y, e)) > 0, there exist continuum many geodesics connecting (X, d) and (Y, e) passing through X(u, v, w).The construction of such geodesics induces a topological embedding of the Hilbert cube into X(u, v, w), and hence X(u, v, w) has infinite topological dimension.In the present paper, as an analogue of this result, we prove that an arbitrary compact metrizable space can be topologically embedded into D(l, a 1 , a 2 , a 3 , a 4 ) and U.
By constructing topological embeddings of an arbitrary compact metrizable space H into D(l, a 1 , a 2 , a 3 , a 4 ) and U which maps given n + 1 points in H into given n + 1 compact metric spaces, we prove the path-connectedness and the infinite-dimensionality of D(l, a 1 , a 2 , a 3 , a 4 ) and U.The existence of these embeddings is based on a construction of a metric on a direct sum space of metric spaces using amalgamations of metrics.
Theorem 1.3.Assume that S = D(l, a 1 , a 2 , a 3 , a 4 ) for some numbers (l, a 1 , a 2 , a 3 , a 4 ) ∈ R or S = U.Let n ∈ Z ≥1 , and H a compact metrizable space.Take mutually distinct n + 1 points {v i } n+1 i=1 in H, and let {(X i , d i )} n+1 i=1 be compact metric spaces in S satisfying that GH((X i , d i ), (X j , d j )) > 0 for all distinct i, j.Then there exists a topological embedding Φ : In [13,Lemma 2.18], it is shown that if D is any one of dim T , dim H , dim P , dim B , dim B , or dim A , then the set of all compact metric space (X, d) such that D(X, d) = ∞ is dense in M. Using the same method, we find that the set D(l, a 1 , a 2 , a 3 , a 4 ) is dense in M.
In this paper, a topological space is said to be infinite-dimensional if its topological dimension is infinite.
As consequences of Theorem 1.3, we obtain the following corollaries.
Corollary 1.5.The set U of all compact ultrametric spaces is pathconnected and infinite-dimensional in M.
Remark 1.1.For a metric space (X, d), the dilation map λ → (X, λd) ) is a geodesic connecting the one-point metric space and (X, d) (see [15, (4) in Proposition 1.4]).From this observation, it follows that (U, GH) is path-connected.Mémoli, Smith and Wan [19] proved that (U, GH) is a geodesic space (see [19,Theorem 7.13]), and proved that (U, GH) is closed and nowhere dense in M (see [19,Proposition 4.17]).Thus, the path-connectedness of (U, GH) have been already known.Our construction of paths is obtained as a by-product of our topological embeddings of an arbitrary compact metrizable space, and this construction is enough to obtain another proof of Qiu's theorem.
1.3.Space of compact ultrametric spaces.We provide applications of Corollary 1.5 to the non-Archimedean Gromov-Hausdorff space.
As a non-Archimedean analogue of the Gromov-Hausdorff distance GH on M, the non-Archimedean Gromov-Hausdorff distance N A on U was defined in [27] (the definition will be presented in Subsection 4.2).
The space (U, N A) is called the non-Archimedean Gromov-Hausdorff space.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 until otherwise stated.From Corollary 1.5, we deduce the relationship between the spaces (U, GH) and (U, N A).
Corollary 1.7.Let I U : (U, GH) → (U, N A) be the identity map of U.Then, I U is not continuous with respect to the topologies induced from GH and N A.
The organization of this paper is as follows: In Section 2, we prepare and explain the basic definitions and statements on metric spaces.The definitions of the five dimensions and GH appearing in Section 1 are given.In Section 3, we define the (m, α)-Cantor ultrametric space, and we give formulas to calculate dimensions of such an ultrametric space.Using that formulas, first we construct spaces stated in Theorem 1.1 in cases of a i ∈ {0, 1, ∞} for all i ∈ {1, 2, 3, 4}.By taking a direct sum of those spaces in specific cases, we prove Theorems 1.1 and 1.2.In Section 4, for each n ∈ Z ≥1 , we construct topological embeddings of an arbitrary compact metrizable space into D(l, a 1 , a 2 , a 3 , a 4 ) and U. We also discuss its applications to the non-Archimedean Gromov-Hausdorff space.
Acknowledgements.The author would like to thank Takumi Yokota for raising questions, for the many stimulating conversations, and for the many helpful comments.The author would also like thank to the referee for helpful suggestions and comments.

Preliminaries
In this section, we prepare and explain the basic concepts and statements on metric spaces.
2.1.Generalities.For k ∈ Z, we denote by Z ≥k the set of all integers greater than or equal to k.For a set S, we denote by Card(S) the cardinality of S. Let (X, d) be a metric space.For x ∈ X and for r ∈ (0, ∞), we denote by B(x, r) the closed ball centered at x with radius r.For a subset A, we denote by δ d (A) the diameter of A.
For two metric spaces (X, d) and (Y, e), we denote by d × ∞ e the ℓ ∞ -product metric defined by (d × ∞ e)((x, y), (u, v)) = d(x, u) ∨ e(y, v).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.
2.2.1.The topological dimension.In this paper, the topological dimension means the covering dimension.For a separable metrizable space, the topological dimension is equal to the large and small inductive dimensions.We refer the readers to [11], [23], [21], and [2] for the details.
2.2.2.The Hausdorff dimension.Let (X, d) be a metric space.For δ ∈ (0, ∞), we denote by F δ (X, d) the set of all subsets of X with diameter smaller than δ.For s ∈ [0, ∞), and δ ∈ (0, ∞), we define the measure H s δ on X as For s ∈ (0, ∞) we define the s-dimensional Hausdorff measure H s on (X, d) as H s (A) = sup δ∈(0,∞) H s δ (A).We denote by dim H (X, d) the Hausdorff dimension of (X, d) defined as

2.2.3.
The packing dimension.Let (X, d) be a metric space.For a subset A of X, and for δ ∈ (0, ∞), we denote by Pa δ (A) the set of all finite or countable sequence {r i } N i=1 (N ∈ Z ≥1 ∪ {∞}) in (0, δ) for which there exists a sequence and a subset A of X, we define the quantity P s δ (A) by We then define the s-dimensional pre-packing measure P s 0 on (X, d) as P s 0 (A) = inf δ∈(0,∞) P s δ , and we define the s-dimensional packing measure P s on (X, d) as We denote by dim P (X, d) the packing dimension of (X, d) defined as Remark 2.1.Our definition of the packing measure is of radius-type.
There is another definition of diameter-type.If we adopt the diametertype definition, then the statement (3) in Theorem 3.4 does not hold true in general.For the difference between two definitions, we refer the readers to [16, Lemma 1.5.7] and [4].These two types of the packing measure are often treated indistinguishably, usually by imposing a regularity condition such as the assumption that the whole metric space is a Euclidean space, or geodesic.
2.2.4.Box dimensions.For a metric space (X, d), and for r ∈ (0, ∞), a subset A of X is said to be an r-net if X = a∈A B(a, r).We denote by N d (X, r) the least cardinality of r-nets of X.We define the upper box dimension dim B (X, d) and the Remark 2.2.In this paper, we do not mainly treat the lower box dimension.
2.2.5.The Assouad dimension.For a metric space (X, d), we denote by dim A (X, d) the Assouad dimension of (X, d) defined by the infimum of all λ ∈ (0, ∞) for which there exists C ∈ (0, ∞) such that for all R, r ∈ (0, ∞) with R > r and for all x ∈ X we have If such λ does not exist, we define dim A (X, d) = ∞.We say that a metric space is doubling if its Assouad dimension is finite.We show some properties of the Assouad dimension.By the definition of the Assouad dimension, we first obtain: Lemma 2.1.Let λ ∈ (0, 1), and N ∈ Z ≥2 .Let (X, d) be a metric space.If every closed ball with radius r can be covered by at most N many closed balls with radius λr, then we have Combining the definitions of Θ X,d (ǫ) and the Assouad dimension, we can verify: Lemma 2.2.Let (X, d) be a metric space.Then, the Assouad dimension dim A (X, d) is equal to the infimum of all λ ∈ (0, ∞) for which there exists C ∈ (0, ∞) such that for all ǫ ∈ (0, 1) we have Θ X,d (ǫ) ≤ Cǫ −λ .Lemma 2.1 implies: Lemma 2.3.For every metric space (X, d), the following statements are equivalent to each other.
(3) For all ǫ ∈ (0, 1), we have The next lemma stats that, to calculate the Assouad dimension, we only need the information of the behavior of Θ X,d (ǫ) on a restricted domain.
Lemmas 2.3 and 2.4 imply the characterization of non-doubling metric spaces: Corollary 2.5.For every metric space (X, d), the following statements are equivalent to each other.
(3) For all ǫ ∈ (0, 1), we have Using the function η X,d , we find a simple formula of the Assouad dimension.
We refer the readers to [6], [7], and [8] for the details of the following.Proposition 2.7.Let (X, d) be a metric space.Let D stand for any one of dim H , dim B , dim P , or dim A .Then the following hold true: (1) For every subset A of X, we have D(A, d) ≤ D(X, d).
(2) Let A, B be subsets of X with A ∪ B = X.Then we have ) Let (Y, e) be a metric space, and f : (X, d) → (Y, e) be an (L, γ)homogeneously bi-Hölder map.Then, we obtain the equality Theorem 2.8.Let (X, d) be a bounded metric space.Then, Proof.The inequality dim T X ≤ dim H (X, d) is due to Szpilrajn [25] (see also [10]).The inequality dim B (X, d) or we can prove it using Proposition 2.6.The proofs of the remaining inequalities are presented in [7, Chapters 2 and 3], which proofs are valid for not only subsets of the Euclidean spaces but also general metric spaces.
For the details of the next proposition, we refer the readers to [11], [23], [21], and [2].Proposition 2.9.Let X and Y be separable metrizable spaces.Then we have dim The proof of the following is presented in [7] and [8].
Proposition 2.10.Let (X, d) and (Y, e) be metric spaces.Then the following statements hold true.
(1) Let D be any one of dim P , dim B , or dim A , then we have ( Remark 2.4.In general, we can not replace dim P with dim H in the righthand side in (2) of Proposition 2.10.Indeed, for all metric spaces (X, d) and (Y, e), we have dim For the details, we refer the readers to [7] and [8].For a set X, a map d : X × X → [0, ∞) is said to be a pseudo-metric if d satisfies the triangle inequality and satisfies that d(x, x) = 0 and d(x, y) = d(y, x) for all x, y ∈ X.If a pseudo-metric d satisfies that d(x, y) = 0 implies x = y, then d is a metric.A pseudo-metric is said to be a pseudo-ultrametric if it satisfies the strong triangle inequality.We denote by PMet(X) (resp.PUMet(X)) the set of all pseudo-metrics (resp.pseudo-ultrametrics) on X.We define a metric D X on PMet(X) by D X (d, e) = sup x,y∈X |d(x, y) − e(x, y)|.Note that D X can take the value ∞.
Proposition 2.12.Let d, e ∈ PMet(X), and assume that e is a metric on X.Then we have Proof.Let p : X → X /d be the canonical projection.Since dis(p) = D X (d, e), the proposition follows from Lemma 2.11.Proposition 2.12 implies: Corollary 2.13.Let T be a topological space all of whose finite subsets are closed.Let X be a set.If a map h : T → PMet(X) is continuous and there exists a finite subset L of T such that h(t) is a metric for all t ∈ T \ L, then the map F : T → M defined by 2.5.Amalgamations of pseudo-metrics.For a topological space X, we denote by Met(X) (resp.UMet(X)) the set of all metrics (resp.all ultrametrics) on X generating the same topology of X.For every n ∈ Z ≥1 , we denote by n the set {1, . . ., n}.In what follows, we consider that the set n is always equipped with the discrete topology.
Since Proposition 3.1 in [14] treats a similar construction, we omit the proofs of the following lemmas.
Lemma 2.14.Let n ∈ Z ≥2 .Let {X i } n i=1 be metrizable spaces and {d i } n i=1 be pseudo-metrics with d i ∈ PMet(X i ).Let r ∈ PMet( n) and p i ∈ X i .We define a symmetric function D on i=1 be ultrametrizable spaces and {d i } n i=1 be pseudo-ultrametrics with d i ∈ PUMet(X i ).Let r ∈ PUMet( n) and p i ∈ X i .We define a symmetric function D on the set

Prescribed dimensions
3.1.Cantor ultrametric spaces.Definition 3.1.Let M be the set of all sequences m = {m i } i∈Z ≥0 of integers with m i ≥ 2 for all i ∈ Z ≥0 .In other words, M = (Z ≥2 ) Z ≥0 .A map α : Z ≥0 → (0, ∞) is said to be a shrinking sequence if α is strictly decreasing and converges to 0. We denote by SH the set of all shrinking sequences.Let S(m) be the set of all maps x from Z ≥0 into Z ≥0 such that x(i) ∈ {0, 1, . . ., m i − 1} for all i ∈ Z ≥0 .We define a valuation v : We also define α ♯ (x, y) = α(v(x, y)), where we put α(∞) = 0. Then α ♯ is an ultrametric on S(m).Notice that (S(m), α ♯ ) is a Cantor space.In this paper, the space (S(m), α ♯ ) is called the (m, α)-Cantor ultrametric space.This space is a generalization of sequentially metrized Cantor spaces defined in the author's paper [12].This construction of Cantor spaces has been utilized in fractal geometry (for example, [16]).
For α ∈ SH, and for m = {m i } i∈Z ≥0 ∈ M, and for k ∈ Z ≥1 , we define the k-shifted shrinking sequence α {k} of α by α {k} (n) = α(n + k), and define k-shifted sequence m {k} = {m From the definition of k-shifted sequences, we deduce the following lemma.
To calculate the Hausdorff dimension and the packing dimension, we use the local dimensions of measures on metric spaces.Let (X, d) be a separable metric space, and µ be a finite Borel measure on X.For every x ∈ X, we define the upper (reps.lower) local dimension dim loc µ(x) (resp.dim loc µ(x)) by The following theorem is well-known.The proofs on the Hausdorff dimension, and the packing dimension are presented in [3, Lemmas 2.1 and 2.2], and [4,Corollary 3.20], respectively.The paper [22, Corollary 2.9] provides its sophisticated version.Proposition 2.3 in the book [6] treats Theorem 3.4 only in the Euclidean setting; however, that proof is also valid for the general setting since the so-called "5r covering lemma" holds true in general metric spaces (see [10,Theorem 1.2]).Theorem 3.4.Let (X, d) be a separable metric space, and µ be a finite Borel measure on X.Let s ∈ [0, ∞).Then, we obtain: (1) If s ≤ dim loc µ(x) for all x ∈ X and µ(X) > 0, then we have (3) If s ≤ dim loc µ(x) for all x ∈ X and µ(X) > 0, then we have s ≤ dim P (X, d).(4) If dim loc µ(x) ≤ s for all x ∈ X, then we have dim P (X, d) ≤ s.Definition 3.2.Let m = {m i } i∈Z ≥0 ∈ M and α ∈ SH.We denote by µ m,α the probability Borel measure on (S(m), α ♯ ) satisfying for all x ∈ S(m) and n ∈ Z ≥0 .Note that µ m,α always exists since it is the countable product of uniform measures on {0, 1, . . ., m i − 1} (see also the argument of "repeated subdivision" in [6] and [7]).

3.2.
Proof of Theorems 1.1 and 1.2.Definition 3.3.For (a 1 , a 2 , a 3 , a 4 ) ∈ L, we say that a metric space (X, d) is of the dimensional type (a 1 , a 2 , a 3 , a 4 ) if we have To prove Theorems 1.1 and 1.2, we construct Cantor ultrametric spaces of the dimensional type (a 1 , a 2 , a 3 , a 4 ) with a i ∈ {0, 1, ∞} for all i ∈ 4 using Propositions 3.3 and 3.5.Definition 3.4.We denote by 2 the sequence in M all of whose entries are equal to 2.
We next construct a Cantor ultrametric space of the dimensional type (0, 0, 1, 1).As noted in Remark 3.1, the packing dimension and the upper box dimension of the (m, α)-Cantor ultrametric space coincide with each other.Thus, a metric space of the dimensional type (0, 0, 1, 1) is not the (m, α)-Cantor ultrametric space for any m ∈ M and any α ∈ SH.We realize it as a subspace of the (m, α)-Cantor ultrametric space using the following theorem due to Mišík-Žáčik [20,Theorem 4].Theorem 3.12.Let (X, d) be an infinite compact metric space.Then for every w ∈ [0, dim B (X, d)], there exist a convergent sequence {a i } i∈Z ≥0 with its limit l ∈ X such that dim B ({l} Lemma 3.13.There exists a Cantor ultrametric space of the dimensional type (0, 0, 1, 1).
We now prove Theorems 1.1 and 1.2.
Proof of Theorem 1.2.Let (l, a 1 , a 2 , a 3 , a 4 ) ∈ R. Based on Theorem 1.1, we may assume that l > 0. In the case of l < ∞, let (M, e) be the metric space ([0, 1] l , d R l ), where d R l is the Euclidean metric.In the case of l = ∞, let (M, e) be the metric space (Q, u), where u ∈ Met(Q).In any case, we notice that the space (M, e) is of the dimensional type (l, l, l, l) and dim T M = l.Due to Theorem 1.1, we can find a Cantor ultrametric space (X, d) of the dimensional type (a 1 , a 2 , a 3 , a 4 ).Put Y = X ⊔ M. Let h be a metric in Met(Y ) with h| X 2 = d and h| ([0,1] l ) 2 = e (see Lemma 2.14).Since dim T M = l, by Proposition 2.9, we conclude that (Y, h) is a metric space as desired.This completes the proof of Theorem 1.2.

Topological embeddings of the Hilbert cube
4.1.Construction of embeddings.We refer to the construction in [13].
The proofs of the metric parts of the following two propositions are presented in [13,Propositions 4.5 and 4.6].The ultrametric parts can be proven by the same method.
Proposition 4.2.The maps ) are isometric to each other.Then we have q = r.
The following proposition occupies the main part of the proof of Theorem 1.3.Proposition 4.4.Let (l, a 1 , a 2 , a 3 , a 4 ) ∈ R, n ∈ Z ≥1 and m ∈ Z ≥2 .Let H be a compact metrizable space and {v i } n+1 i=1 be n + 1 points in H. Put H × = H \ { v i | i = 1, . . ., n + 1 }.Let {(X i , d i )} n+1 i=1 be a sequence of compact metric spaces satisfying that GH((X i , d i ), (X j , d j )) > 0 for all distinct i, j.Then there exists a continuous map F : H × m → D(l, a 1 , a 2 , a 3 , a 4 ) such that (1) for all i ∈ n + 1 and j ∈ m we have F (v i , j) = (X i , d i ); (2) for all (s, i), (t, j) ∈ H × × m with (s, i) = (t, j), we have F (s, i) = F (t, j).
Put Z = n+2 i=1 Y i , and take r ∈ Met( n + 2) and p i ∈ Y i .For each (s, k) ∈ H × m, we define a symmetric function D s,k on Z by D s,k (x, y) = E i,s,k (x, y) if x, y ∈ Y i ; E i,s,k (x, p i ) + ξ(s)r(i, j) + E j,s,k (p j , y) if x ∈ Y i and y ∈ Y j .Lemma 2.14 implies that D s,k is a pseudo-metric on Z for all (s, k) ∈ H × m.We notice that D s,k is a metric if and only if s ∈ H × .We also notice that for all i ∈ n + 1 and k ∈ m, the quotient metric space Z /D v i ,k , [D v i ,k ] is isometric to (X i , d i ).Combining Propositions 2.7, 2.9, and 2.10, and Lemma 4.1, we conclude that (Z, D s,k ) ∈ D(l, a 1 , a 2 , a 4 , a 4 ) for all (s, k) ∈ H × × m.
We define a map F : H × m → D(l, a 1 , a 2 , a 3 , a 4 ) by Next we prove the condition (2).For a metric space (S, h), we denote by CI(S, h) the closure of the set of all isolated point of (S, h).Note that if metric spaces (S, h) and (S ′ , h ′ ) are isometric to each other, then so are CI(S, h) and CI(S ′ , h ′ ).Since (P, v) has no isolated points, so does Y i for all i ∈ n + 1.Then, by the definitions of Ω, and D s,k , the space CI(Z, D s,k ) is isometric to (Ω, ξ(s) • u[τ (s, k)]) for all (s, k) ∈ H × × m.Since CI is isometrically invariant, and since τ is injective, Proposition 4.3 implies that the condition (2) is satisfied.This finishes the proof.
Similarly to Proposition 4.4, we obtain its analogue for ultrametrics.Proposition 4.5.Fix n ∈ Z ≥0 .Let H be a compact metrizable space and {v i } n+1 i=1 be n+1 points in H. Put H × = H \{ v i | i = 1, . . ., n+1 }.Let m ∈ Z ≥2 .Let {(X i , d i )} n+1 i=1 be a sequence of compact ultrametric spaces with GH((X i , d i ), (X j , d j )) > 0 for all distinct i, j.Then there exists a continuous map F : H × m → U such that (1) for all i ∈ n + 1 and j ∈ m we have F (v i , j) = (X i , d i );

2. 4 .
The Gromov-Hausdorff distance.For a metric space (Z, h), and for subsets A, B of Z, we denote by HD(A, B; Z, h) the Hausdorff distance of A and B in Z.For metric spaces (X, d) and (Y, e), the Gromov-Hausdorff distance GH((X, d), (Y, e)) between (X, d) and (Y, e) is defined as the infimum of all values HD(i(X), j(Y ); Z, h), where (Z, h) is a metric space, and i : X → Z and j : Y → Z are isometric embeddings.Let f : X → Y be a map between metric spaces (X, d) and (Y, e).We define the distortion dis(f ) of f bydis(f ) = sup x,y∈X |d(x, y) − e(f (x), f (y))|.The following is deduced from[1, Corollary 7.3.28].Lemma 2.11.Let (X, d) and (Y, e) be compact metric spaces, and f : X → Y be a surjective map.Then GH((X, d), (Y, e)) ≤ 2 dis(f ).
Let d ∈ PMet(X).We denote by X /d the quotient set by the relation ∼ d defined by x ∼ d y ⇐⇒ d(x, y) = 0. We also denote by [x] d the equivalence class of x by ∼ d .We define a metric [d] on X /d by [ for some i ∈ n + 1; (Z, D s,k ) otherwise.Then the condition (1) is satisfied.By the definition of D s,k , and by Proposition 4.2, the map W : H × m → PMet(Z) defined by W (s, k) = D s,k is continuous.Therefore, Corollary 2.13 guarantees the continuity of the map F .