arXiv subjects

Yoshito Ishiki

Publications and source records attributed to Yoshito Ishiki.

At least 19 recordsLinked to original sources

arXiv2026

Algebraically independent distances and rigid metrics

We study metrics whose distances on distinct two-point subsets are algebraically independent over the rationals. We prove that every compatible metric on a strongly zero-dimensional metrizable space of cardinality at most continuum can be uniformly approximated by compatible metrics with this property. If the space is completely metrizable, the approximating metrics can also be chosen complete. For every $σ$-compact metrizable space, the metrics with algebraically independent distances form a $G_δ$ set in the uniform topology. We also study rigid metrics, whose only bijective self-isometry is the identity. On every locally compact Polish space, rigid proper metrics form a $G_δ$ set among proper compatible metrics. Using a theorem of Niemiec, we obtain uniform density of rigid metrics on compact metrizable spaces with at least three points. Passing to compact completions then yields rigid approximations of every totally bounded compatible metric on any space with at least three points.

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arXiv2026

The topology of Gromov--Hausdorff space

We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. This paper is divided into four parts. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.

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arXiv2026

Borelness of Moduli Spaces of Metrics Implies Separability

Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the supremum-metric topology. We first prove that if Z is a discrete space of cardinality aleph-one, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is sigma-compact. We also establish non-Archimedean analogues for spaces of ultrametrics.

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arXiv2026

A non-Archimedean Arens--Eells isometric embedding theorem on valued fields

In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$.

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arXiv2026

Absolute Borel Complexity of Moduli Spaces of Ultrametrics

Let $X$ be an ultrametrizable space. We study the space of bounded compatible ultrametrics on $X$, equipped with its natural non-Archimedean distance. For every positive integer level, we prove that additive absolute Borel complexity of this moduli space implies multiplicative absolute Borel complexity of $X$ at the same level, and conversely. We also prove that an ultrametrizable space is a countable union of locally compact subspaces if and only if it is a countable union of closed subsets in every completion induced by a bounded compatible ultrametric. As a consequence, this moduli space is completely metrizable exactly when $X$ is a countable union of compact subsets.

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arXiv2026

Algebraic structures on non-Archimedean Urysohn universal metric spaces

We investigate valued-field structures on Urysohn universal ultrametric spaces. We introduce $p$-adic Levi--Civita fields as subfields of $p$-adic Hahn fields and treat them together with ordinary Levi--Civita fields. For a subgroup $G$ of $\mathbb{R}$ containing $\mathbb{Z}$ and a countably infinite perfect field $k$, the corresponding Levi--Civita valued field is isometric to the $R$-Urysohn universal ultrametric space, where $R=\{0\}\cup\{η^{-g}\mid g\in G\}$. Thus these spaces admit field structures extending prescribed prime valued fields, including $\mathbb{Q}$ with the trivial valuation and the $p$-adic fields $\mathbb{Q}_{p}$. We also prove that complete valued fields with infinite residue fields are haloed, and hence universal for separable ultrametric spaces with corresponding distance sets. In the separable case, such a valued field is itself isometric to the corresponding Urysohn space. Examples include $\mathbb{C}_{p}$, the completion of the maximal unramified extension of $\mathbb{Q}_{p}$, Laurent series fields, and completions of their algebraic closures. Finally, for a countably infinite perfect residue field, the corresponding full Hahn-type valued field is a Urysohn universal ultrametric space exactly when its value group is order-isomorphic to $\mathbb{Z}$.

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arXiv2026

An interpolation of metrics and spaces of metrics

As a generalization of Hausdorff's extension theorem of metrics, we prove an interpolation theorem of a family of metrics defined on closed subsets of metrizable spaces. As an application, we investigate typicality of subsets of moduli spaces of metrics. We observe that various sets of all metrics with properties appearing in metric geometry are dense intersections of countable open subsets in spaces of metrics on metrizable spaces. For instance, our study is applicable to the set of all non-doubling metrics and the set of all non-uniformly disconnected metrics.

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arXiv2026

Strongly rigid metrics in spaces of metrics

A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of space of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metics. Moreover, if the space is the union of countable compact subspaces, then that set is comeager. As its consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.

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arXiv2026

A factorization of metric spaces

We first prove that for every metrizable space $X$, for every closed subset $F$ whose complement is zero-dimensional, the space $X$ can be embedded into a product space of the closed subset $F$ and a metrizable zero-dimensional space as a closed subset. Using this theorem, we next show the existence of extensors of metrics and ultrametrics, which preserve properties of metrics such as the completeness, the properness, being an ultrametrics, its fractal dimensions, and large scale structures. This result contains some of the author's extension theorems of ultrametrics.

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arXiv2024

On isometric universality of spaces of metrics

A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first prove that the space of bounded metrics on $Z$ is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if $Z$ is an infinite discrete space, then the space of metrics on $Z$ is universal for all separable metric spaces. As a corollary of our results, if $Z$ is non-compact, or uncountable and compact, then the space of metrics on $Z$ is universal for all compact metric spaces. In addition, if $Z$ is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on $Z$.

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arXiv2024

An isometric extensor of metrics

In this paper, for a metrizable space $Z$, we consider the space of metrics that generate the same topology of $Z$, and that space of metrics is equipped with the supremum metrics. For a metrizable space $X$ and a closed subset $A$ of it, we construct a map $E$ from the space of metrics on $A$ into the space of metrics on $X$ such that $E$ is an extension of metrics and preserves the supremum metrics between metrics.

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arXiv2024

Characterizations of Urysohn universal ultrametric spaces

In this paper, using the existence of infinite equidistant subsets of closed balls, we characterize the injectivity of ultrametric spaces for finite ultrametric spaces, which also gives a characterization of the Urysohn universal ultrametric spaces. As an application, we find that the operations of the Cartesian product and the hyperspaces preserve the structures of the Urysohn universal ultrametric spaces. Namely, let $(X, d)$ be the Urysohn universal ultrametric space. Then we show that $(X\times X, d\times d)$ is isometric to $(X, d)$. Next we prove that the hyperspace consisting of all non-empty compact subsets of $(X, d)$ and symmetric products of $(X, d)$ are isometric to $(X, d)$. We also establish that every complete ultrametric space injective for finite ultrametric space contains a subspace isometric to $(X, d)$.

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arXiv2024

Spaces of metrics are Baire

For a metrizable space, we consider the space of all metrics generating the same topology of the metrizable space, and this space of metrics is equipped with the supremum metric. In this paper, for every metrizable space, we establish that the space of metrics on the metrizable space is Baire. We also show that the set of all complete metrics is comeager in the space of metrics. Moreover, we investigate non--Archimedean analogues of these results.

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arXiv2023

Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces

Urysohn constructed a separable complete universal metric space homogeneous for all finite subspaces, which is today called the Urysohn universal metric space. Some authors have recently investigated an ultrametric analogue of this space. The isometry problem of such ultrametric spaces is our main subject in this paper. We first introduce the new notion of petaloid ultrametric spaces, which is intended to be a standard class of non-separable Urysohn universal ultrametric spaces. Next we prove that all petaloid spaces are isometric to each other and homogeneous for all finite subspaces (and compact subspaces). Moreover, we show that the following spaces are petaloid, and hence they are isometric to each other and homogeneous: (1) The space of all continuous functions, whose images contain the zero, from the Cantor set into the space of non-negative real numbers equipped with the nearly discrete topology, (2) the space of all continuous ultrametrics on a zero-dimensional infinite compact metrizable space, (3) the non-Archimedean Gromov--Hausdorff space, and (4) the space of all maps from the set of non-negative real numbers into the set of natural numbers whose supports are finite or decreasing sequences convergent to the zero.

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arXiv2023

On comeager sets of metrics whose ranges are disconnected

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ is equipped with the supremum distance. In this paper, for every strongly zero-dimensional metrizable space $X$, we prove that the set of all metrics whose ranges are closed totally disconnected subsets of the line is a dense $G_δ$ subspace in $\mathrm{Met}(X)$. As its application, we show that some sets of universal metrics are meager in spaces of metrics.

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arXiv2023

Constructions of Urysohn universal ultrametric spaces

In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of non-negative real numbers equipped with the nearly discrete topology. As a consequence, the whole function space is Urysohn universal, which can be considered as a non-Archimedean analog of Banach--Mazur theorem. As a more application, we prove that the space of all continuous pseudo-ultrametrics on a zero-dimensional compact Hausdorff space with an accumulation point is a Urysohn universal ultrametric space. This result can be considered as a variant of Wan's construction of Urysohn universal ultrametric space via the Gromov--Hausdorff ultrametric space.

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arXiv2022

Extending proper metrics

We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $σ$-compact locally compact Hausdorff spaces. As its application, we prove an extension theorem of proper metrics, which states that if $X$ is a $σ$-compact locally compact space, $A$ is a closed subset of $X$, and $d$ is a proper metric on $A$ that generates the same topology of $A$, then there exists a proper metric on $X$ such that $D$ generates the same topology of $X$ and $D|_{A^{2}}=d$. Moreover, if $A$ is a proper retraction, we can choose $D$ so that $(A, d)$ is quasi-isometric to $(X, D)$. We also show analogues of theorems explained above for ultrametric spaces.

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arXiv2022

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 dimension, 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 infinite 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.

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