On certain star versions of a \(\mathrm{U}_{\mathrm{fin}}\)-type selection principle (Q6592035)

In this paper the authors consider the star versions of the selection principle \(\text{U}_{\text{fin}}(\mathcal{O}, \Omega)\), namely \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\), \(\text{SS}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\) and \(^{\ast}\text{U}_{\text{fin}}(\mathcal{O}, \Omega)\). They improve some results by showing that every star-Lindelöf (respectively, strongly star-Lindelöf) space of cardinality less than \(\mathfrak{d}\) is \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\) (respectively, \(\text{SS}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\)).\N\NThe authors prove the following results.\N\N\textbf{Theorem 3.5.}\N\N(1) If \(X\) is a star-Lindelöf space, then for any subset \(Y\) with cardinality less than \(\mathfrak{d}\) \(X\) satisfies \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega_{Y})\).\N\N(2) If \(X\) is a strongly star-Lindelöf space, then for any subset \(Y\) with cardinality less than \(\mathfrak{d}\) \(X\) satisfies \(\text{SS}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega_{Y})\).\N\N\textbf{Theorem 3.14.}\N\N(1) Let \(X\) be Lindelöf. If \(X\) is a union of less than \(\mathfrak{d}\) \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Gamma)\) spaces, then \(X\) is \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\).\N\N(2) Let \(X\) be star-Lindelöf. If \(X\) is a union of less than \(\mathfrak{d}\) \(\text{U}_{\text{fin}}(\mathcal{O}, \Gamma)\) spaces, then \(X\) is \(\text{U}_{\text{fin}}^{\ast}(\mathcal{O}, \Omega)\).\N\N\textbf{Theorem 4.8.} Let \(X\) be a star-Lindelöf regular P-space. If \(Y\) is an infinite closed and discrete subset of \(X\), then \(|Y|<\text{cof}(\text{Fin}(w(X))^{\mathbb{N}})\).\N\N\textbf{Theorem 5.5.} If \(X\) is a space such that ONE does not have a winning strategy in the game \(\text{G}_{\text{ufin}}^{\ast}(\mathcal{O}, \Omega)\) on \(X\), then each countable large cover of \(X\) is star-weakly groupable.