Remarks on star-K-Menger spaces (Q907810)

Recall that a space \(X\) is star-K-Menger if for each sequence \(\mathcal{U}_1, \mathcal{U}_2, \dots \) of open covers of \(X\) there exists a sequence \(K_1, K_2, \dots \) of compact subsets of \(X\) such that \(\{St(K_n, \mathcal{U}_n): n\in \mathbb{N}\}\) covers \(X\). A space \(X\) is star-Menger if for each sequence \(\mathcal{U}_1, \mathcal{U}_2, \dots \) of open covers of \(X\) there exists a sequence \(\mathcal{V}_1, \mathcal{V}_2, \dots \) such that for each \(n\in \mathbb{N}\), \(\mathcal{V}_n\) is a finite subset of \(\mathcal{U}_n\) and \(\bigcup_{n\in \mathbb{N}}\{St(V, \mathcal{U}_n): V\in \mathcal{V}_n\}\) covers \(X\). Clearly star-K-Menger spaces are star-Menger spaces. In the paper under review, the author investigates star-K-Menger spaces. The main results are as follows: there exists a Hausdorff star-Menger space which is not star-K-Menger; for a \(T_1\) star-K-Menger space \(X\), its Alexandroff duplicate~\(A(X)\) is star-K-Menger if and only if \(e(X)<\omega_1\); there exists a Tychonoff star-K-Menger space \(X\) such that \(A(X)\) is not star-K-Menger; the star-K-Menger property is hereditary with respect to open and closed subspaces, but it is not hereditary with respect to regular-closed \(G_{\delta}\)-subspaces. The author asks whether there exists a regular (or Tychonoff) star-Menger space which is not star-K-Menger.