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A metric space $X$ is defined to be an absolute Lipschitz extensor for compacta if each Lipschitz map $f:K\to K$ defined on a compact subset $K\subset X$ extends to a Lipschitz map $\bar f: X\to X$. …

If $(E_t)_{t\in T}$ is a partition of a metric space $(X,d)$ into parallel compact subsets, then the metric $d$ induces a quotient metric $\hat d$ on the index set $T$. This metric is defined by $\hat …

The answer to this problem is affirmative (at least for covers). Definition. A family $\mathcal C$ of subsets of a topological space $X$ is called $\bullet$ lower semicontinuous if for any open set …

For $C=\omega^\omega$ the answer is affirmative: Theorem 1. Let $X$ be a metric space, $Y$ be separable metric space, and $F,G\subset X\times Y$ be closed sets such that $\inf\{d(x,y):x\in F,\;y\in …

By a Tarski plane I understand a set $X$ endowed with a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying all Tarski axioms except fo …

It seems that the conjecture (H2) can be confirmed with help of the recent result of Hofmann and Kramer (http://arxiv.org/pdf/1301.5114.pdf) who proved that for a compact topological group $G$ and a c …

Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map …

I would suggest the following axioms. The length in a metric space $X$ is a function $\ell:c(X)\to[0,+\infty]$ defined on the family $c(X)$ of all connected compact subsets of $X$ that satisfies the …

Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of …

Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y …

By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. Unfortunat …

This problem was motivated by the MO problems: "Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?". L …

The answer is given by the Lebesgue's Covering Theorem (numbered as 1.8.20 in Engleking's book "Theory of dimensions: finite and infinite"): If $\mathcal F$ is a finite closed cover of the $n$-cube $I …

It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles. Question. Is an analogo …

Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??) implies …