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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 …

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 …

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 …

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 …

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 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 $xy$. …

The answer to this question is affirmative. Proposition. Let $p:X\to Y$ be a proper bijective map from a Hausdorff topological space $X$ onto a $T_1$-space $Y$. Then for every continuous map $f:K\to …

Let us say that a topological space $X$ is spherically completely metrizable if the topology of $X$ is generated by a spherically complete metric. Theorem. Every closed subspace $X$ of the countable …

$\newcommand\abs[1]{\lvert#1\rvert}$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, …

I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$: Theorem. Let $n$ be positive integer number. A metric space $X$ is isom …

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 …

Let $f:M\to Y$ be a continuous proper bijective map from a metrizable space $M$ onto a $T_1$-space $Y$. The properness of $f$ means that for every compact subspace $K\subseteq Y$ the preimage $f^{-1}[ …

The space $X$ constructed in Theorem 1.5 of this preprint has the required properties. This space contains a non-closed compact metrizable subspace $K$, so is not weakly Hausdorff. On the other hand, …

Not necessarily: consider the compactly generated space $Y=\mathbb R^\infty=\varinjlim \mathbb R^n$, which is the direct limit of Euclidean spaces. Then for the countable discrete space $X=\omega$ th …

The second question also has the negative answer: Take any sequence $z=(z_n)_{n\in\omega}\in[0,1]^\omega$ with $f(z)=0$. On the Hilbert cube $[0,1]^\omega$, consider the metric $d(x,y)=\max_{n\in\omeg …