Related papers: Pseudocompact group topologies with no infinite co…

The \emph{Bohr compactification} of an arbitrary topological group $G$ is defined as the group compactification $(bG,b)$ with the following universal property: for every continuous homomorphism $h$ from $G$ into a compact group $K$ there is…

Let $G $ be a group of cardinality $\kappa>\aleph_0 $ endowed with a topology $\tau $ such that $|U|=\kappa$ for every non-empty $U\in\tau$ and $\tau$ has a base of cardinality $\kappa$. We prove that $G$ could be factorized $G=AB$ (i.e.…

A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued…

A non-trivial topological group is called \emph{$d$-independent} if for every subgroup of cardinality less than the continuum there exists a countable dense subgroup intersecting it trivially. This notion was introduced by M\'arquez and…

We prove that finitely generated higher dimensional Kleinian groups with small critical exponent are always convex-cocompact. Along the way, we also prove some geometric properties for any complete pinched negatively curved manifold with…

We study when a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor space $X$ can be given a non-discrete totally disconnected locally compact (t.d.l.c.) topology and give a criterion for the alternating…

A precompact group topology $\tau$ on an abelian group $G$ is called {\em single sequence characterized} (for short, {\em ss-characterized}) if there is a sequence $\mathbf{u}= (u_n)$ in $G$ such that $\tau$ is the finest precompact group…

Two non-discrete Hausdorff group topologies $\tau, \delta$ on a group $G$ are called {\it transversal} if the least upper bound $\tau\vee \delta$ of $\tau$ and $\delta$ is the discrete topology. In this paper, we discuss the existence of…

We generalize the notion of an approximate indicator for a closed subgroup $H$ of a locally compact group $G$ introduced by Aristov, Runde, and Spronk and extend their characterization of the existence of such nets in terms of the…

Recently it was shown that the category of cocommutative Hopf algebras over an arbitrary field $\Bbbk$ is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the…

A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological…

Let G be an abelian topological group. The symbol \hat{G} denotes the group of all continuous characters \chi : G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that \chi(E) \subseteq…

We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let $X$ be a second countable locally compact Abelian group containing no subgroups topologically…

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description…

In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct…

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural…

We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

We denote by C_p(X,G) the group of all continuous functions from a space X to a topological group G endowed with the topology of pointwise convergence. We say that spaces X and Y are G-equivalent provided that the topological groups…