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...

A polyadic space is a Hausdorff continuous image of some power of the one-point compactification of a discrete space. We prove a Ramsey-like property for polyadic spaces which for Boolean spaces can be stated as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint. One corollary is that ${\left(\alpha \kappa \right)}^{\omega }$ is not a universal preimage for uniform Eberlein compact spaces of weight at most κ, thus answering a question of Y. Benyamini, M. Rudin and M. Wage....

A question raised in R. G. Woods' paper has a simple solution.

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 ${\bigcup }_{fin}(𝒪,𝒪)$.

Following the introduction of separability in frames ([2]) we investigate further properties of this notion and establish some consequences of the Urysohn metrization theorem for frames that are frame counterparts of corresponding results in spaces. In particular we also show that regular subframes of compact metrizable frames are metrizable.

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...

Normal spaces are characterized in terms of an insertion type theorem, which implies the Katětov-Tong theorem. The proof actually provides a simple necessary and sufficient condition for the insertion of an ordered pair of lower and upper semicontinuous functions between two comparable real-valued functions. As a consequence of the latter, we obtain a characterization of completely normal spaces by real-valued functions.