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Is the following metric characterization of the real line true (and known)? A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real …

This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of Bessaga and Pelczynski, Chigogidze or Sakai. Factor Theorem. For any Polish abs …

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable partition …

The chromatic number $\chi(X)$ of a topological space $X$ is related to the separation dimension $t(X)$ introduced and studied by Steinke. The separation dimension $t(X)$ is defined inductively: $\ …

The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and …

The following theorem (or its corollary) implies negative answer to the original question. Theorem. For any point $x$ of a metric space $(X,d)$ the set $R_x:=\{r>0:cl(B(x,r))\ne \bar B(x,r)\}$ has …

First let us fix the terminology. The space (1) is known in General Topology as the Golomb space. More precisely, the Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers, endowed wit …

Following the answer of Uri Bader, we can show that $Y$ is a retract of the real line, so can be identified with a closed convex subset of $\mathbb R$. Without loss of generality we can assume that $0 …

[Edit, Dec 6, 2019] I have a pleasure to inform that this problem was finally resolved in affirmative by T.Banakh, D.Spirito and S.Turek who proved the following Theorem. The Golomb space is topologi …

Probably, the discrete $\{0,1\}$ is not the counterexample Dominic van der Zypen expected to see :) A more elaborate CH-example of a AFPP but not strongly rigid space was constructed by van Mill: T …

This is not an answer to the original question of Willie Wong but a partial answer to the following related Problem. Recognize pairs of topological spaces $X,Y$ for which every Darboux injection $f\c …

Theorem. For a topological space $X$ the following conditions are equivalent: 1) for any continuous map $f:X\times S^2\to\mathbb R^2$ there exists a point $s\in S^2$ such that $f(x,s)=f(x,-s)$ …

I noticed that this question still has no accepted answer and all existing answers are rather long. It seems that the answer can be easily obtained using some results of infinite-dimensional topology, …

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a …

For a general audience it can be interesting to know that uniform spaces are just one of two opposite generalizations of metric spaces. Measuring distances with the help of metric, we can be intereste …