Remarks on weakly linearly Lindelöf spaces. (English) Zbl 1422.54028
Authors’ abstract: Recall that a space \(X\) is weakly linearly Lindelöf if for any family \(\mathcal{U}\) of non-empty open subsets of \(X\) of regular uncountable cardinality \(\kappa\), there exists a point \(x\in X\) such that every neighborhood of \(x\) meets \(\kappa\)-many elements of \(\mathcal{U}\) . In this paper, we show that:
(1) If \(X\) is a weakly linearly Lindelöf space and \(\mathcal{U}\) is an open cover of \(X\), then for the open cover \(\{ St^2(x,\mathcal{U}):\) \(x \in X\}\) of \(X\), there exists a countable subset \(A\subset X\) such that \(\overline{St^2(A,\mathcal{U})}=X\);
(2) Every weakly linearly Lindelöf normal meta-Lindelöf space is weakly Lindelöf;
(3) If \(X\) is a first countable regular space, then \(\mathcal{M}\)\((X)\) (generated by the Moore Machine) is weakly linearly Lindelöf if and only if \(X\) is weakly linearly Lindelöf;
(4) Every product of a weakly linearly Lindelöf space and a space of countable spread (or a separable space) is weakly linearly Lindelöf;
(5) If a subspace \(X \subset \omega_1^{\omega}\) is weakly linearly Lindelöf, then \(X\) is second countable (and hence, metrizable);
(6) If \(X\) is a weakly linearly Lindelöf Baire space with a rank 2-diagonal such that \(we(X)\leq \omega_1\), then \(|X|\leq c\) ;
(7) The space X is cellular-WLL if and only if it is weakly linearly Lindelöf.
(1) If \(X\) is a weakly linearly Lindelöf space and \(\mathcal{U}\) is an open cover of \(X\), then for the open cover \(\{ St^2(x,\mathcal{U}):\) \(x \in X\}\) of \(X\), there exists a countable subset \(A\subset X\) such that \(\overline{St^2(A,\mathcal{U})}=X\);
(2) Every weakly linearly Lindelöf normal meta-Lindelöf space is weakly Lindelöf;
(3) If \(X\) is a first countable regular space, then \(\mathcal{M}\)\((X)\) (generated by the Moore Machine) is weakly linearly Lindelöf if and only if \(X\) is weakly linearly Lindelöf;
(4) Every product of a weakly linearly Lindelöf space and a space of countable spread (or a separable space) is weakly linearly Lindelöf;
(5) If a subspace \(X \subset \omega_1^{\omega}\) is weakly linearly Lindelöf, then \(X\) is second countable (and hence, metrizable);
(6) If \(X\) is a weakly linearly Lindelöf Baire space with a rank 2-diagonal such that \(we(X)\leq \omega_1\), then \(|X|\leq c\) ;
(7) The space X is cellular-WLL if and only if it is weakly linearly Lindelöf.
Reviewer: Maximilian Ganster (Graz)
MSC:
| 54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
| 54E35 | Metric spaces, metrizability |
Keywords:
weakly linearly Lindelöf; weakly Lindelöf; Moore spaces; countable spread; Baire space; rank 2-diagonal; cellular-WLLReferences:
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