Convergence in compacta and linear Lindelöfness. (English) Zbl 0937.54022

Arhangel’skii, A. V.; Buzyakova, R. Z.

Commentat. Math. Univ. Carol. 39, No. 1, 159-166 (1998).

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

Keywords:

point of complete accumulation; linearly Lindelöf space; local compactness; first countability; \(\kappa \)-accessible diagonal