We give a construction of Wallman-type realcompactifications of a frame $L$ by considering regular sub $\sigma$-frames the join of which generates $L$. In particular, we show that the largest such regular sub $\sigma$-frame gives rise to the universal realcompactification of $L$.

We give a characterization of a paracompact $\Sigma$-space to have a $G_{\delta }$-diagonal in terms of three rectangular covers of $X^2\setminus \Delta$. Moreover, we show that a local property and a global property of a space $X$ are given by the orthocompactness of $(X\times \beta X)\setminus \Delta$.

We say that a cardinal function $\phi$ reflects an infinite cardinal $\kappa$, if given a topological space $X$ with $\phi \left(X\right)\ge \kappa$, there exists $Y\in {\left[X\right]}^{\le \kappa }$ with $\phi \left(Y\right)\ge \kappa$. We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences with...

We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition,...

Arhangel’skiĭ defines in [Topology Appl. 70 (1996), 87–99], as one of various notions on relative topological properties, strong normality of $A$ in $X$ for a subspace $A$ of a topological space $X$, and shows that this is equivalent to normality of $X_A$, where $X_A$ denotes the space obtained from $X$ by making each point of $X\setminus A$ isolated. In this paper we investigate for a space $X$, its subspace $A$ and a space $Y$ the normality of the product $X_A\times Y$ in connection with the normality of ${(X\times Y)}_{(A\times Y)}$. The cases for paracompactness, more...

We continue the study of remainders of metrizable spaces, expanding and applying results obtained in [Fund. Math. 215 (2011)]. Some new facts are established. In particular, the closure of any countable subset in the remainder of a metrizable space is a Lindelöf p-space. Hence, if a remainder of a metrizable space is separable, then this remainder is a Lindelöf p-space. If the density of a remainder Y of a metrizable space does not exceed $2^{\omega }$, then Y is a Lindelöf Σ-space. We also show that many of...

We prove the following statements: (1) every Tychonoff linked-Lindelöf (centered-Lindelöf, star countable) space can be represented as a closed subspace in a Tychonoff pseudocompact absolutely star countable space; (2) every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed G δ-subspace in a Hausdorff (regular, Tychonoff) absolutely star countable space; (3) there exists a pseudocompact absolutely star countable Tychonoff space having a regular closed subspace which...

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming $2^{{\aleph }_0}=2^{{\aleph }_1}$, there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

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 $X=\mathrm{St}(A,𝒰)$, where $\mathrm{St}(A,𝒰)=\bigcup \{U\in 𝒰:U\cap A\ne \varnothing \}.$ In this paper, we study the relationships of star $P$ properties for $P\in \{\mathrm{Lindel}ö\mathrm{f},\mathrm{compact},\mathrm{countablycompact}\}$ in pseudocompact spaces by giving some examples.

A space X is star-Hurewicz if for each sequence (𝒰ₙ: n ∈ ℕ) of open covers of X there exists a sequence (𝓥ₙ: n ∈ ℕ) such that for each n, 𝓥ₙ is a finite subset of 𝒰ₙ, and for each x ∈ X, x ∈ St(⋃ 𝓥ₙ,𝒰ₙ) for all but finitely many n. We investigate the relationship between star-Hurewicz spaces and related spaces, and also study topological properties of star-Hurewicz spaces.

A space $X$ is strongly star-Menger if for each sequence $({𝒰}_n:n\in \mathbb{N})$ of open covers of $X$, there exists a sequence $(K_n:n\in N)$ of finite subsets of $X$ such that $\{St(K_n,{𝒰}_n):n\in \mathbb{N}\}$ is an open cover of $X$. In this paper, we investigate the relationship between strongly star-Menger spaces and related spaces, and also study topological properties of strongly star-Menger spaces.

We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger than the countable chain condition. It is proved that Martin's Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same can be forced...