The class of $s$-spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf $p$-spaces, metrizable spaces with the weight $\le 2^{\omega }$, but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that $s$-spaces are in a duality with Lindelöf $\Sigma$-spaces: $X$ is an $s$-space if and only if some (every) remainder of $X$ in a compactification is a Lindelöf $\Sigma$-space [Arhangel’skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. 220 (2013),...

This paper presents a new consistent example of a relatively normal subspace which is not Tychonoff.

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group....

Let $P$ be a topological property. A space $X$ is said to be star P if whenever $𝒰$ is an open cover of $X$, there exists a subspace $A\subseteq X$ with property $P$ such that $X=St(A,𝒰)$. In this note, we construct a Tychonoff pseudocompact SCE-space which is not star Lindelöf, which gives a negative answer to a question of Rojas-Sánchez and Tamariz-Mascarúa.

In this note we first give a summary that on property of a remainder of a non-locally compact topological group $G$ in a compactification $bG$ makes the remainder and the topological group $G$ all separable and metrizable. If a non-locally compact topological group $G$ has a compactification $bG$ such that the remainder $bG\setminus G$ of $G$ belongs to $𝒫$, then $G$ and $bG\setminus G$ are separable and metrizable, where $𝒫$ is a class of spaces which satisfies the following conditions: (1) if $X\in 𝒫$, then every compact subset of the space $X$ is a...

We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by...

We prove for a subspace $Y$ of a $T_1$-space $X$, $Y$ is (strictly) Aull-paracompact in $X$ and $Y$ is Hausdorff in $X$ if and only if $Y$ is strongly star-normal in $X$. This result provides affirmative answers to questions of A.V. Arhangel’skii–I.Ju. Gordienko [3] and of A.V. Arhangel’skii [2].