Every semi-stratifiable space or strong $\Sigma$-space has a $\sigma$-cushioned (mod$k$)-network. In this paper it is showed that every space with a $\sigma$-cushioned (mod$k$)-network is a D-space, which is a common generalization of some results about D-spaces.

Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_{\delta }$-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\left\{x\right\}=\bigcap \left\{g\right(x,n):n\in \mathbb{N}\}$ and $g(x,n+1)\subseteq g(x,n)$ for each $n\in \mathbb{N}$. (2) If a...

We show that a space is MCP (monotone countable paracompact) if and only if it has property $(*)$, introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally countably compact closed mappings, from which it follows that both strongly Fréchet spaces and q-space closed images of MCP spaces are MCP. Some results on...

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.

A topological space $X$ is said to be star Lindelöf if for any open cover $𝒰$ of $X$ there is a Lindelöf subspace $A\subset X$ such that $St(A,𝒰)=X$. The “extent” $e\left(X\right)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. We prove that under $V=L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $\mathrm{MA}+\neg \mathrm{CH}$, which shows that a star Lindelöf, first countable and normal space may not have countable extent.

We prove that every Tychonoff strongly monotonically monolithic star countable space is Lindelöf, which solves a question posed by O.T. Alas et al. We also use this result to generalize a metrization theorem for strongly monotonically monolithic spaces. At the end of this paper, we study the extent of star countable spaces with k-in-countable bases, k ∈ ℤ.

In this note, we show that if for any transitive neighborhood assignment $\phi$ for $X$ there is a point-countable refinement $ℱ$ such that for any non-closed subset $A$ of $X$ there is some $V\in ℱ$ such that $|V\cap A|\ge \omega$, then $X$ is transitively $D$. As a corollary, if $X$ is a sequential space and has a point-countable $wc{s}^{*}$-network then $X$ is transitively $D$, and hence if $X$ is a Hausdorff $k$-space and has a point-countable $k$-network, then $X$ is transitively $D$. We prove that if $X$ is a countably compact sequential space and has a point-countable...

In this paper, a simple proof is given for the following theorem due to Blair [7], Blair-Hager [8] and Hager-Johnson [12]: A Tychonoff space $X$ is $z$-embedded in every larger Tychonoff space if and only if $X$ is almost compact or Lindelöf. We also give a simple proof of a recent theorem of Bella-Yaschenko [6] on absolute embeddings.

Given a topological property (or a class) $𝒫$, the class ${𝒫}^{*}$ dual to $𝒫$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\{O_x:x\in X\}$ there is $Y\subset X$ with $Y\in 𝒫$ and $\bigcup \{O_x:x\in Y\}=X$. The spaces from ${𝒫}^{*}$ are called dually $𝒫$. We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define $D$-spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space...

In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal $𝔤$ is a lower bound of the additivity number of the $\sigma$-ideal generated by Menger subspaces of the Baire space, and under $𝔲<𝔤$ every subset $X$ of the real line with the property $Split(\Lambda ,\Lambda )$ is Hurewicz, and thus it is consistent with ZFC that the property $Split(\Lambda ,\Lambda )$ is preserved by unions of less than $𝔟$ subsets of the real line.

Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property ${\bigcup }_{fin}(𝒪,T^{*})$ provided $(𝔲<𝔤)$, and every space with the property ${\bigcup }_{fin}(𝒪,T^{*})$ is Hurewicz provided $({Depth}^{+}\left({\left[\omega \right]}^{{\aleph }_0}\right)\le 𝔟)$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text{P}$ and $\text{Q}$ [do not] coincide, where $\text{P}$ and $\text{Q}$ run over ${\bigcup }_{fin}(𝒪,\Gamma )$, ${\bigcup }_{fin}(𝒪,T)$, ${\bigcup }_{fin}(𝒪,T^{*})$, ${\bigcup }_{fin}(𝒪,\Omega )$, and ${\bigcup }_{fin}(𝒪,𝒪)$.

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le 2^{}$, a topological group G such that $G^{\gamma }$ is countably compact for all cardinals γ < α, but $G^{\alpha }$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has remained...