## 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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