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Let $Z=\{0\}\cup\{\pm\frac1n\}_{n\in\mathbb N}$ be the sequence that converges to zero from both sides. Consider the contractible continuum $$A=(Z\times[-1,1]\times\{0\})\cup([-1,1]\times\{0\}\times\{ …

I am looking for a characterization of topological spaces $X,Y$ for which the function space $C_k(X,Y)$ is Baire. Here $C_k(X,Y)$ is the space of continuous functions from $X$ to $Y$, endowed with th …

Let $X$ be a topological space. Let $[X]^{<\omega}$ be the space of non-empty finite subsets of $X$, endowed with the Vietoris topology. For a natural number $n$ the subspace $$[X]^{\le n}:=\{A\in[X]^ …

A topological space $X$ is called $\bullet$ sequential if for each non-closed subset $A\subset X$ there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$; $\bu …

Problem. Does every compact countable space contain a non-trivial convergent sequence? This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed wi …

It is well-known that each uncountable compact metrizable space $X$ contains a homeomorphic copy of the Cantor cube $\{0,1\}^\omega$. What about copies of Cantor cubes of larger weight? Problem. Does …

For a cardinal $\kappa$ let $\Box^\kappa\mathbb R$ be the box-product of $\kappa$-many lines and $\boxdot^\kappa\mathbb R:=\{x\in\Box^\kappa:|\{\alpha\in\kappa:x(\alpha)\ne 0\}<\omega\}$ be the $\sigm …

This question was motivated by my answer to this MO question, which asked about the characterization of spaces that belong to the smallest class of topological spaces that is closed under taking subsp …

Let us recall that a regular topological space is semi-stratifiable if each point $x\in X$ has a countable family of neighborhoods $(U_n(x))_{n\in\omega}$ such that each closed subset $F\subset X$ is …

A topological space $X$ is defined to be $\bullet$ cometrizable if $X$ admits a weaker metrizable topology $\tau$ such that for every point $x\in X$ and a neighborhood $U_x\subseteq X$ of $x$ there e …

Let us recall that a symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that for every $x,y\in X$ the following two conditions are satisfied: $d(x,y)=0$ if and only if $x=y$; $d( …

This question is inspired by this (still unanswered) MO-post. A function $f:X\to Y$ between topological spaces is called strongly rigid if every continuous self-map $h:X\to X$ with $f\circ h=f$ is the …

I am looking for the name and notation of the following separation axiom , temporarily denoted by $T_i$ (where $i=\sqrt{-1}$ is the imaginary unit): Axiom $T_i$: For any point $x$ of a topological sp …

Problem. Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$. Does $D$ contain a non-trivial convergent sequence in …

Let us recall that a symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that for every $x,y\in X$ the following two conditions are satisfied: $\bullet$ $d(x,y)=0$ if and only if $x …