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A space $X$ is {\it star-K-Menger} if for each sequence $(\mathcal U_n:n\in\Bbb N)$ of open covers of $X$ there exists a sequence $(K_n:n\in N)$ of compact subsets of $X$ such that $\{St(K_n,\mathcal U_n):n\in\Bbb N\}$ is an open cover of $X$. In this paper, we construct an example of a Hausdorff star-Menger space that is not star-K-Menger which gives an answer to a question of Song [12], and continue to investigate topological properties of star-K-Menger spaces.

A star covering property which is equivalent to countable compactness for regular spaces and weaker than countable compactness for Hausdorff spaces is introduced and considered. Various kinds of irregularity of topological spaces are discussed Quaestiones Mathematicae 30(2007), 407–415

[EN] We introduce two new notions of topological spaces called a countably starcompact space and a countably absolutely countably compact (= countably acc) space. We clarify the relations between these spaces and other related spaces and investigate topological properties of countably starcompact spaces and countably acc spaces. Some examples showing the limitations of our results are also given., The author is supported by NSFC project 10571081 and by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No 07KJB110055)

It is shown that from any first-countable, zero-dimensional, locally compact, non-DFCC space X with the property that every nonempty open set has π-weight c one can construct a pseudocompact space Y+ that is not strongly 2-star-Lindelof. Such a space Y+ would therefore be 1. (1) 2-starcompact, T3 but not strongly 2-starcompact; 2. (2) 2-star-Lindelof, T3 but not strongly 2-star-Lindelof. A space satisfying (1) has been constructed using CH and a Moore space with a σ-locally countable base satisfying (2) is known. The examples generated here are easier than these two spaces, require no set theory beyond ZFC and make the distinctions (1) and (2) simultaneously.