Related papers: Isometric embeddings and continuous maps onto the …

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…

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…

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum…

We prove that if a separable metrizable $X$ is a union of two disjoint 0-dimensional sets $E$, $F$, $E$ is absolutely $G_{\delta}$ and $F$ is absolutely $F_{\sigma\delta}$ then there is a closed embedding $h$ into the union of countable…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…

In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form…

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…

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of…

Let M be a complete metric space. It is proved that if the space or scalar-valued bounded continuous functions on M admits an isometric shift, then M is separable.

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and…

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and…

We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) =…

We investigate when a map on a selfadjoint operator space $E$ is an embedding, i.e., when its unitisation in the sense of Werner is completely isometric. Combining with results of Russell, of Ng, and of Dessi, the second and the last…

We show that, for a separable and complete metric space $M$, the Lipschitz-free space $\mathcal F(M)$ embeds linearly and almost-isometrically into $\ell_1$ if and only if $M$ is a subset of an $\mathbb R$-tree with length measure 0.…

In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio…

We study the existence of Riemannian metrics with zero topological entropy on a closed manifold M with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to M in…

The aim of the paper is to prove that if $M$ is a metrizable manifold modelled on a Hilbert space of dimension $\alpha \geq \aleph_0$ and $F$ is its $\sigma$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than…

We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite…

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible…