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A space X is star-K-Hurewicz if for each sequence (Un : n ? N) of open covers of X there exists a sequence (Kn : n ? N) of compact subsets of X such that for each x ? X, x ? St(Kn,Un) for all but finitely many n. In this paper, we investigate the relationship between star-K-Hurewicz spaces and related spaces by giving some examples, and also study topological properties of star-K-Hurewicz spaces.

A space $X$ is {\it weakly Hurewicz} if for each sequence $(\mathcal U_n:n\in\Bbb N)$ of open covers of $X$, there are a dense subset $Y\subseteq X$ and finite subfamilies $\mathcal V_n\subseteq \mathcal U_n (n\in \Bbb N)$ such that for every point of $Y$ is contained in $\bigcup\mathcal V_n$ for all but finitely many $n$. In this paper, we investigate the relationship between Hurewicz spaces and weakly Hurewicz spaces, and also study topological properties of weakly Hurewicz spaces.

A space X is said to be neighborhood star-Lindelof if for every open cover U of X there exists a countable subset A of X such that for every open O ⊇ A, X = St(O, U). In this paper, we continue to investigate the relationship between neighborhood star-Lindel¨ of spaces and related spaces, and study topological properties of neighborhood star-Lindel¨ of spaces in the classes of normal and pseudocompact spaces.

In this paper, we show the following statements: (1) There exists a Tychonoff pseudocompact, star-Hurewicz space having a regular-closed subspace which is not star-Hurewicz; (2) Assuming ω1 < b, there exists a Tychonoff strongly star-Hurewicz (hence star-Hurewicz) space having a regular-closed Gδ-subspace which is not star-Hurewicz (hence not strongly star-Hurewicz); (3) An open Fσ-subset of a strongly star-Hurewicz space is strongly star-Hurewicz.

A space X is strongly star-Hurewicz if for each sequence (Un : n ∈ N) of open covers of X there exists a sequence (An : n ∈ N) of finite subsets of X such that for each x ∈ X, x ∈ St(An; Un) for all but finitely many n. In this paper, we continue to investigate topological properties of strongly star-Hurewicz spaces.