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Convergence in compacta and linear Lindelöfness. (English) Zbl 0937.54022
Summary: Let \(X\) be a compact Hausdorff space with a point \(x\) such that \(X\smallsetminus \{ x\}\) is linearly Lindelöf. Is then \(X\) first countable at \(x\)? What if this is true for every \(x\) in \(X\)? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when \(X\) is, in addition, \(\omega \)-monolithic. We also prove that if \(X\) is compact, Hausdorff, and \(X\smallsetminus \{ x\}\) is strongly discretely Lindelöf, for every \(x\) in \(X\), then \(X\) is first countable. An example of a linearly Lindelöf hereditarily realcompact non-Lindelöf space is constructed. Some intriguing open problems are formulated.

MSC:
54F99 Special properties of topological spaces
54E35 Metric spaces, metrizability
54D30 Compactness
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