Characterization Of Metric Spaces Research Articles

Quasisymmetric Koebe uniformization with weak metric doubling measures

We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by Merenkov and Wildrick.

Duality of Lipschitz classes on compact metric spaces

Let $$(X, d)$$ be a compact metric space and $${\mathrm {lip}}(X, d)$$ the closed subspace of the Lipschitz functions space $${\mathrm {Lip}}(X, d)$$ whose elements f satisfy the “vanishing” property $$\begin{aligned} \lim _{d(x,y) \rightarrow 0} \frac{\vert f(x) - f(y) \vert }{d(x,y)} = 0. \end{aligned}$$We consider characterizations of metric spaces $$(X, d)$$ for which the completion of the space $${\mathscr {M}}(X)$$ of Borel measures on $$(X, d)$$ is isomorphic to $${\mathrm {lip}}(X, d)^*.$$

McShane’s Extension Theorem Revisited

A fundamental extension theorem of McShane states that a bounded real-valued uniformly continuous function defined on a nonempty subset A of a metric space 〈X, d〉 can be extended to a uniformly continuous function on the entire space. In the first half of this note, we obtain McShane’s Extension Theorem from the simpler fact that a real-valued Lipschitz function defined on a nonempty subset of the space has a Lipschitz constant preserving extension to the entire space. In the second half of the note, we use this theorem to give an elementary proof of the equivalence of the most important characterizations of metric spaces in which the real-valued uniformly continuous functions form a ring. These characterizations of such a basic property, due to Cabello-Sanchez and separately Bouziad and Sukhacheva, are remarkably recent.

Some characterization of metric spaces in which every bounded set is conditionally compact

Some characterization of metric spaces in which every bounded set is conditionally compact

Finitely chainable and totally bounded metric spaces: Equivalent characterizations

A metric space (X,d) is called finitely chainable if for every ϵ>0, there are finitely many points p1,p2,...,pr in X and a positive integer m such that every point of X can be joined with some pj, 1≤j≤r by an ϵ-chain of length m. In 1958, Atsuji proved: a metric space (X,d) is finitely chainable if and only if every real-valued uniformly continuous function on (X,d) is bounded. In this paper, we study twenty-five equivalent characterizations of finitely chainable metric spaces, out of which three are entirely new. Here we would like to mention that this study essentially turns the first part of this paper into a sort of an expository research article. A totally bounded metric space is finitely chainable. In order to have a better perception of the difference between total boundedness and finite chainability, several new equivalent characterizations of totally bounded metric spaces are also studied. Moreover, two topological characterizations of metric spaces admitting compatible finitely chainable metrics are given.

Characterization of metric spaces whose free space is isometric to $\ell_1$

We characterize metric spaces whose Lipschitz free space is isometric to $\ell_1$. In particular, we show that the Lipschitz free space over an ultrametric space is not isometric to $\ell_1(\Gamma)$ for any set $\Gamma$. We give a lower bound for the Banach-Mazur distance in the finite case.

Between compactness and completeness

Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement.

On weak-open π-images of metric spaces

In this paper, we give some characterizations of metric spaces under weak-open π-mappings, which prove that a space is g-developable (or Cauchy) if and only if it is a weak-open π-image of a metric space.

On π‐s‐images of metric spaces

We establish the characterizations of metric spaces under compact‐covering (resp., pseudo‐sequence‐covering, sequence‐covering) π‐s‐maps by means of cfp‐covers (resp., sfp‐covers, cs‐covers) and σ‐strong networks.

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Čech-complete spaces. We also show that the absolute Borel class of $X$ is determined by the uniform structure of the space of continuous functions $C_{p}(X)$; however the case of absolute $G_{\delta }$ metric spaces is still open. More precisely, we prove that, for metrizable spaces $X$ and $Y$, if $\Phi : C_{p}(X) \rightarrow C_{p}(Y)$ is a uniformly continuous surjection and $X$ is an absolute Borel set of multiplicative (resp., additive) class $\alpha$, $\alpha >1$, then $Y$ is also an absolute Borel set of the same class. This result is new even if $\Phi$ is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Čech-completeness of a metric space $X$ is determined by the linear structure of $C_{p}(X)$.

More about metric spaces on which continuous functions are uniformly continuous

An Atsuji space is a metric space X such that each continuous function form X to an arbitrary metric space Y is uniformly continuous. We here present (i) characterizations of metric spaces with Atsuji completions; (ii) Cantor-type theorems for Atsuji spaces; (iii) a fixed point theorem for self-maps of an Atsuji space.

Characterizations of metric spaces by the use of their midsets: intervals

Characterizations of metric spaces by the use of their midsets: intervals