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Let 𝑋 be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set. Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\ …

Among separable metrizable spaces: Cantor set is the unique compact zero-dimensional space without isolated points. $\mathbb Q$ is the unique countable space without isolated points $\mathbb R \se …

Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$ Can you always find a closed nowhere dens …

Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $ …

Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed quot …

I would like a simple description of a dense subset of $\mathbb R$ which is homeomorphic to $\mathbb Q\times \mathbb P$. Preferably the description will be of an algebraic nature, and perhaps the set …

An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space …

This question is about the Erdős spaces: $\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and $\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\},$ w …

Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact neighb …

Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology. Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0. Let $Y$ be the set of …

The pseudo-arc is the unique hereditarily indecomposable chainable continuum. The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense endpoi …

Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let \begin{align} \Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\ \Lambda^2(X)&=\Lambda(\Lambda(X) …

A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense. Does either of the Erdös spaces $\m …

A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$ is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of clo …

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish topologica …