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It is well-known that each abelian group admits an injective homomorphism to some compact topological group (for example to its Bohr compactification). Is the same fact true for solvable groups? Ques …

A simple example of sequential topological groups $G,H$ with non-sequential product $G\times H$: $G=\mathbb R^\omega$ and $H=\mathbb R^\infty=\lim \mathbb R^n$ be the direct limit of finite-dimensiona …

The answer to both problems (1 and 2) is negative: the Polish group $G=\mathbb Z^\omega$ contains a dense meager Borel subgroup $H$ (which can be written as the difference $H=A\setminus B$ of two $F_\ …

The Gleason-Montgomery theorem on the local compactness of locally path-connected finite-dimensional topological groups was generalized by Banakh and Zdomskyy (http://topology.auburn.edu/tp/reprints/v …

Problem. Let $\lambda$ be the standard product measure on the Hilbert cube $[-\frac12,\frac12]^\omega$ and $A,B$ be two $\lambda$-positive Borel subsets of $[-\frac12,\frac12]^\omega$. Is it true tha …

Definition. A topological group $G$ is called autoseparable if there exists a countable subset $S\subset G$ and a sequence $(f_n)_{n\in\omega}$ of automorphisms of $G$ such that for any neighborhood $ …

Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subse …

A topological group $G$ is called $\bullet$ minimal if it admits no strictly weaker Hausdorff group topology; $\bullet$ completely minimal if it is Raikov-complete in each weaker Hausdorff group to …

For a topological group $X$ by $Aut(X)$ denote the group of topological isomorphisms $h:X\to X$. If $X$ is compact then the compact-open topology turns $Aut(X)$ into an $\omega$-narrow topological gro …

A topological group is called minimal if it admits no strictly weaker Hausdorff group topology. By Prodanov-Stoyanov Theorem, a complete Abelian topological group is minimal if and only if it is comp …

A topological group $G$ is called $\omega$-$\mathit{narrow}$ if for each non-empty open set $U\subset G$ there exists a countable subset $C\subset G$ such that $G=CU=UC$; $\mathit{feathered}$ if …

The phenomemnon of minimality is well-studied in the realm of topological groups. Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a topolo …

This question is motivated by this MO-problem asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ admit a self-homeomorphism with dense orbits of points. The affirmative answer would follow …

A topological group $G$ is defined to be $\bullet$ precompact if for any neighborhood $U\subseteq G$ of the unit there exists a finite subset $F\subseteq G$ such that $G=UF$; $\bullet$ narrow if for …

Jan Pachl has informed me that the answer to this problem is affirmative and can be derived from the following helpful fact, proved in Lemma 3.31 of his book "Uniform spaces and measures". I also reme …