The quasi-Rothberger property of linearly ordered spaces
Recall that a space \(X\) is said to be Rothberger if for each sequence \(\{\mathscr{U}_n: n\in \mathbb{N}\}\) of open covers of \(X\), there is a \(U_n\in \mathscr{U}_n\) for each \(n\in \mathbb{N}\) such that \(\{U_n: n\in \mathbb{N}\}\) covers \(X\); a space \(X\) is said to be quasi-Rothberger if for each closed set \(F\subset X\) and each sequence \(\{\mathscr{U}_n: n\in \mathbb{N}\}\) of open covers of \(F\) by sets open in \(X\), there is a \(U_n\in \mathscr{U}_n\) for each \(n\in \mathbb{N}\) such that \(F\subset \overline{\bigcup_{n\in \mathbb{N}}U_n}\); a space \(X\) is said to be quasi-Lindelöf if for each closed set \(F\subset X\) and each open cover \(\mathscr{U}\) of \(F\) by open sets of \(X\), there is a countable family \(\{U_n: n\in \mathbb{N}\}\) of \(\mathscr{U}\) such that \(F\subset \overline{\bigcup_{n\in \mathbb{N}}U_n}\). Clearly, the quasi-Rothberger property is weaker than the Rothberger property and stronger than the quasi-Lindelöf property; the Lindelöf property is weaker than the Rothberger property and stronger than the quasi-Lindelöf property. There are no implications between the quasi-Rothberger property and the Lindelöf property (see Example 1.2 and Example 1.3 of the paper). In the paper under review the author studies the quasi-Rothberger property and obtains mainly the following results. (1) For a metrizable space \(X\), the quasi-Rothberger property, the quasi-Lindelöf property and separability are equivalent. (2) Let \((X, <) \) be a linearly ordered space of cardinality \(\alpha\) with \(cf(\alpha)>\omega\); then the following are equivalent: (a) \(X\) is quasi-Rothberger; (b) \(X\) is quasi-Lindelöf; (c) \(e(X)=\omega\) and each closed well-ordered (resp., closed anti-well-ordered) subset of \(X\) of order type \(\alpha\) with \(cf(\alpha)>\omega\) has a largest (resp., a smallest) element, where for a well-ordered set \((X,<)\), the converse order \(>\) of \(<\) is said to be anti-well-ordered of \((x,<)\). The author poses a natural question: Find a ``inner'' characterization in a linearly ordered space \(X\) such that \(X\) is quasi-Rothberger.
scientific article; zbMATH DE number 6924612