### On $\mathcal{K}$-starcompact spaces.

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A space X is absolutely strongly star-Hurewicz if for each sequence (Un :n ∈ℕ/ of open covers of X and each dense subset D of X, there exists a sequence (Fn :n ∈ℕ/ of finite subsets of D such that for each x ∈X, x ∈St(Fn; Un) for all but finitely many n. In this paper, we investigate the relationships between absolutely strongly star-Hurewicz spaces and related spaces, and also study topological properties of absolutely strongly star-Hurewicz spaces.

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...

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 $\mathcal{L}$-starcompact if for every open cover $\mathcal{U}$ of $X,$ there exists a Lindelöf subset $L$ of $X$ such that $\mathrm{S}t(L,\mathcal{U})=X.$ We clarify the relations between $\mathcal{L}$-starcompact spaces and other related spaces and investigate topological properties of $\mathcal{L}$-starcompact spaces. A question of Hiremath is answered.

In this paper, we prove the following two statements: (1) There exists a discretely absolutely star-Lindelöf Tychonoff space having a regular-closed subspace which is not CCC-Lindelöf. (2) Every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff) absolutely star-Lindelöf space as a closed ${G}_{\delta}$ subspace.

A space $X$ is if for every open cover $\mathcal{U}$ of $X$ and every dense subset $D$ of $X$, there exists a countable subset $F$ of $D$ such that $F$ is discrete closed in $X$ and $St(F,\mathcal{U})=X$, where $St(F,\mathcal{U})=\bigcup \{U\in \mathcal{U}:U\cap F\ne \varnothing \}$. We show that every Hausdorff star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed subspace.

A space $X$ is strongly star-Menger if for each sequence $({\mathcal{U}}_{n}:n\in \mathbb{N})$ of open covers of $X$, there exists a sequence $({K}_{n}:n\in N)$ of finite subsets of $X$ such that $\{St({K}_{n},{\mathcal{U}}_{n}):n\in \mathbb{N}\}$ 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.

In this paper, we prove the following statements: (1) For any cardinal $\kappa $, there exists a Tychonoff star-Lindelöf space $X$ such that $a\left(X\right)\ge \kappa $. (2) There is a Tychonoff discretely star-Lindelöf space $X$ such that $aa\left(X\right)$ does not exist. (3) For any cardinal $\kappa $, there exists a Tychonoff pseudocompact $\sigma $-starcompact space $X$ such that $st\text{-}l\left(X\right)\ge \kappa $.

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 $\mathcal{U}$ is an open cover of $X$, there exists a subspace $A\subseteq X$ with property $P$ such that $X=St(A,\mathcal{U})$. 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 paper, we generalize Vaughan's and Bonanzinga's results on absolute countable compactness of product spaces and give an example of a separable, countably compact, topological group which is not absolutely countably compact. The example answers questions of Matveev [8, Question 1] and Vaughan [9, Question (1)].

In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa $, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X)\ge \kappa $, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf...

Let $P$ be a topological property. A space $X$ is said to be star $P$ if whenever $\mathcal{U}$ is an open cover of $X$, there exists a subspace $A\subseteq X$ with property $P$ such that $X=\mathrm{St}(A,\mathcal{U})$, where $\mathrm{St}(A,\mathcal{U})=\bigcup \{U\in \mathcal{U}:U\cap A\ne \varnothing \}.$ In this paper, we study the relationships of star $P$ properties for $P\in \{\mathrm{Lindel}\xf6\mathrm{f},\mathrm{compact},\mathrm{countablycompact}\}$ in pseudocompact spaces by giving some examples.

A space $X$ is if for every open cover $\mathcal{U}$ of $X,$ there exists a countably compact subset $C$ of $X$ such that $\mathrm{St}(C,\mathcal{U})=X.$ In this paper we investigate the relations between $\mathcal{C}$-starcompact spaces and other related spaces, and also study topological properties of $\mathcal{C}$-starcompact spaces.

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