Related papers: Embeddable properties of metric $\sigma$-discrete …
A reduction of properties (invariants) of compact sets of real numbers to properties of countable orders is presented here. Discussed here is also an embedding property of some compact sets that are called t$\mathbb R$-sets. Among others,…
A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable…
A well-known theorem of Assouad states that metric spaces satisfying the doubling property can be snowflaked and bi-Lipschitz embedded into Euclidean spaces. Due to the invariance of many geometric properties under bi-Lipschitz maps, this…
A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric…
We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions;…
Given an uncountable subset $\mathcal Y$ of a nonseparable Banach space, is there an uncountable $\mathcal Z\subseteq \mathcal Y$ such that the distances between any two distinct points of $\mathcal Z$ are more or less the same? If an…
In this paper a construction of a metrizable zero-dimensional CDH space $X$ such that $X^2$ has exactly $\mathfrak{c}$ countable dense subsets is provided. Furthermore, it is shown that the space can be constructed consistently co-analytic.…
The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an…
We construct a class of metric spaces whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\omega+k$ for any $k\in\mathbb{N}$, where $\omega$ is the smallest infinite ordinal number and a metric…
For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…
This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms…
Given a metric space $(X,d)$, a set $S\subseteq X$ is called a $k$-\emph{metric generator} for $X$ if any pair of different points of $X$ is distinguished by at least $k$ elements of $S$. A $k$-\emph{metric basis} is a $k$-metric generator…
We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…
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…
The embedding of a given point set with non-crystallographic symmetry into higher-dimensional space is reviewed, with special emphasis on the Minkowski embedding known from number theory. This is a natural choice that does not require an a…
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic…
We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of…
The main purpose of the paper is to find some expansion properties of locally finite metric spaces which do not embed coarsely into a Hilbert space. The obtained result is used to show that infinite locally finite graphs excluding a minor…
We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric…