General Topology
Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation…
Let $\FP(X)$ be the free paratopological group on a topological space $X$. For $n\in \N$, denote by $\FP_n(X)$ the subset of $\FP(X)$ consisting of all words of reduced length at most $n$, and by $i_n$ the natural mapping from $(X\oplus…
The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of…
In the paper we describe the structure of $\mathscr{AH}$-completions and $\mathscr{H}$-completions of the discrete semilattices $(\mathbb{N},\min)$ and $(\mathbb{N},\max)$. We give an example of an $\mathscr{H}$-complete topological…
For a topological space it is well-known that the associated closure and interior operators provide equivalent descriptions of set-theoretic topology; but it is not generally true in other categories, consequently it makes sense to define…
A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. Contrarily to charges, maxitive measures often have a density. We show that maxitive measures…
The recent extensions of domain theory have proved particularly efficient to study lattice-valued maxitive measures, when the target lattice is continuous. Maxitive measures are defined analogously to classical measures with the supremum…
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which…
We study the relationship between generalizations of $P$-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense $G_\delta$ have dense (non-empty) intersection. In particular, we prove that every dense and every…
Noetherian type and Noetherian $\pi$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the…
The notion of Hausdorff number of a topological space is first introduced in \cite{bonan}, with the main objective of using this notion to obtain generalizations of some known bounds for cardinality of topological spaces. Here we consider…
A common generalization for two of the main streams of cardinality inequalities is developed; each stream derives from the famous inequality established by A.V. Arhangel'ski\u{\i} in 1969 for Hausdorff spaces. At the end of one stream is…
This article extends the recent study of the cardinal functions $F_{\theta}$ and $t_{\theta}$ for H-closed Urysohn spaces and the research of I. Bandlov and V.I. Ponomarev on tightness type of absolutes. In particular, basic results are…
In this article, starting with results from a paper by Cammaroto and Ko\v{c}inac in 1993, we prove that $t_{\theta}(X) = t_{\theta}(X_{s}) = t(X_{s}) = F_{\theta}(X) = F_{\theta}(X_{s}) = F(X_{s})$ for every H-closed and Urysohn space and…
The research in this paper is a continuation of the investigation of the cardinality of the $\theta$-closed hull of subsets of spaces. This research obtains new upper bounds of the cardinality of the $\theta$-closed hull of subsets using…
A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{-1}(A)$ of each nowhere dense subset $A\subset Y$ is nowhere dense in $X$. We prove that a normal functor $F:Comp\to Comp$ is skeletal (which means that $F$…
We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between w-spectra with surjective limit projections.
Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar…
A subset of a Polish space $X$ is called universally small if it belongs to each ccc $\sigma$-ideal with Borel base on $X$. Under CH in each uncountable Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$ such…
We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for…