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A small amendment of the result of Hidehiko mentioned in the answer of Moishe Kohan: Theorem 2.2 in this paper implies that every continuum-connected subgroup of $\mathbb R^n$ is a (closed) linear sub …

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia …

A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$. We say that a topological …

The superextension $\lambda X$ of a compact Hausdorff space $X$ is the space of maximal linked systems of closed subsets of $X$, endowed with the Vietoris topology inherited from the double hyperspace …

According to this SE-post, a Lawvere metric on a set $X$ is a function $d:X\times X\to[0,+\infty]$ satisfying two axioms: $d(x,x)=0$ and $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$. Then the fo …

$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$. Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that $d(x,y)=d_X(x, …

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 …

It is well-known that each topological group $G$ carries (at least) four natural uniformities: the left uniformity $\mathcal L$, generated by the base $\{\{(x,y)\in G\times G:y\in xU\}:U\in\mathcal …

I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces. By a linear space I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of s …

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 …

The affirmative answer to this problem follows from a general result on extension of graph metrics. In the following definitions, the unordered pair $\{x,y\}$ of two elements $x,y$ is denoted by $xy$. …

It seems that Gutik's hedgehog is a required counterexample. I recall that Gutik's hedgehog is the set $$X=\{(0,0)\}\cup\{(\tfrac1n,0):n\in\mathbb N\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\}$$ e …

Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb …

Open sets with the required property are exactly functionally open sets. Let us recall that a subset $U$ of a topological space $X$ is functionally open in $X$ if $U=f^{-1}[V]$ for some continuous fun …

The following modification of the Vietoris topology seems to satisfy the requirements. Let $\tau$ be the topology on $2^X$ consisting of the sets $\mathcal U\subset 2^X$ such that for every closed set …