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point of complete accumulation; linearly Lindelöf space; local compactness; first countability; $\kappa $-accessible diagonal
Let $X$ be a compact Hausdorff space with a point $x$ such that $X\setminus \{ x\}$ 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 $X\setminus \{ x\}$ is strongly discretely Lindelöf, for every $x$ in $X$, then $X$ is first countable. An example of linearly Lindelöf hereditarily realcompact non-Lindelöf space is constructed. Some intriguing open problems are formulated.
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