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For every discrete group $G$, the Stone-Čech compactification $\beta G$ of $G$ has a natural structure of a compact right topological semigroup. An ultrafilter $p\in G^{*}$, where $G^{*}=\beta G\setminus G$, is called right cancellable if, given any $q,r\in G^{*}$, $qp=rp$ implies $q=r$. For every right cancellable ultrafilter $p\in G^{*}$, we denote by $G\left(p\right)$ the group $G$ endowed with the strongest left invariant topology in which $p$ converges to the identity of $G$. For any countable group $G$ and any right cancellable ultrafilters $p,q\in G^{*}$, we show that $G\left(p\right)$ is homeomorphic to $G\left(q\right)$ if and only if...

We show that, under CH, the corona of a countable ultrametric space is homeomorphic to ${\omega }^{*}$. As a corollary, we get the same statements for the Higson’s corona of a proper ultrametric space and the space of ends of a countable locally finite group.

Answering recent question of A.V. Arhangel'skii we construct in ZFC an extremally disconnected semitopological group with continuous inverse having no open Abelian subgroups.

A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. We introduce and study a new cardinal invariant of a ballean, the extraresolvability, which is an asymptotic reflection of the corresponding invariant of a topological space.

Given a discrete group $G$, we consider the set $ℒ\left(G\right)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $ℒ\left(G\right)$ into the Cantor cube ${\{0,1\}}^G$. We show that the cellularity $c\left(ℒ\left(G\right)\right)\le {\aleph }_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c\left(ℒ\right(G\left)\right)=\tau$.