We characterize clean elements of $ℛ\left(L\right)$ and show that $\alpha \in ℛ\left(L\right)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that ${𝔠}_L\left(\mathrm{coz}(\alpha -\mathbf{1})\right)\subseteq U\subseteq {𝔬}_L\left(\mathrm{coz}\left(\alpha \right)\right)$. Also, we prove that $ℛ\left(L\right)$ is clean if and only if $ℛ\left(L\right)$ has a clean prime ideal. Then, according to the results about $ℛ\left(L\right),$ we immediately get results about ${𝒞}_c\left(L\right).$

By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and...

In this paper we continue with the study of paracompact maps introduced in [1]. We give two external characterizations for paracompact maps including a characterization analogous to The Tamano Theorem in the category $𝒯OP$ (of topological spaces and continuous maps as morphisms). A necessary and sufficient condition for the Tychonoff product of a closed map and a compact map to be closed is also given.

We prove resolvability and maximal resolvability of topological spaces having countable tightness with some additional properties. For this purpose, we introduce some new versions of countable tightness. We also construct a couple of examples of irresolvable spaces.

We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = supκ : $2^{\kappa }$ ⊂ X. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered...

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 consider the spaces called $Seq\left(u_t\right)$, constructed on the set $Seq$ of all finite sequences of natural numbers using ultrafilters $u_t$ to define the topology. For such spaces, we discuss continuity, homogeneity, and rigidity. We prove that $S\left(u_t\right)$ is homogeneous if and only if all the ultrafilters $u_t$ have the same Rudin-Keisler type. We proved that a space of Louveau, and in certain cases, a space of Sirota, are homeomorphic to $Seq\left(p\right)$ (i.e., $u_t=p$ for all $t\in Seq$). It follows that for a Ramsey ultrafilter $p$, $Seq\left(p\right)$ is a topological group....