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...

In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $\left|D\right|=d\left(X\right)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma$-space $X$ and hence $X^{\omega }$ is $d$-separable. We give an example of a countably compact space $X$ such that $X^{\omega }$ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta =\{(x,x):x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta$ is a retract of...

If $X$ is a space, then the $we\left(X\right)$ of $X$ is the cardinal $min\{\alpha :$ If $𝒰$ is an open cover of $X$, then there exists $A\subseteq X$ such that $\left|A\right|=\alpha$ and $St(A,𝒰)=X\}$. In this note, we show that if $X$ is a normal space such that $\left|X\right|=𝔠$ and $we\left(X\right)=\omega$, then $X$ does not have a closed discrete subset of cardinality $𝔠$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $𝔠$, even if the condition that $we\left(X\right)=\omega$ is replaced by the stronger condition that $X$ is separable.