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...

A subset $A$ of a Hausdorff space $X$ is called an $\omega$H-set in $X$ if for every open family $𝒰$ in $X$ such that $A\subset \bigcup 𝒰$ there exists a countable subfamily $𝒱$ of $𝒰$ such that $A\subset \bigcup \{\overline{V}:V\in 𝒱\}$. In this paper we introduce a new cardinal function $t_{s\theta }$ and show that $\left|A\right|\le 2^{t_{s\theta }\left(X\right){\psi }_c\left(X\right)}$ for every $\omega$H-set $A$ of a Hausdorff space $X$.

In this paper two new cardinal functions are introduced and investigated. In particular the following two theorems are proved: (i) If $X$ is a functionally Hausdorff space then $\left|X\right|\le 2^{fs\left(X\right){\psi }_{\tau }\left(X\right)}$; (ii) Let $X$ be a functionally Hausdorff space with $fs\left(X\right)\le \kappa$. Then there is a subset $S$ of $X$ such that $\left|S\right|\le 2^{\kappa }$ and $X=\bigcup \{c{l}_{\tau \theta }\left(A\right):A\in {\left[S\right]}^{\le \kappa }\}$.