Abstract:Let $X$ be a compact Hausdorff space with a point $x$ such that $X\setminus \{ x\}$ is linearly Lindel\"of. 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 Lindel\"of, for every $x$ in $X$, then $X$ is first countable. An example of linearly Lindel\"of hereditarily realcompact non-Lindel\"of space is constructed. Some intriguing open problems are formulated.
Keywords: point of complete accumulation, linearly Lindel\"of space, local compactness, first countability, $\kappa $-accessible diagonal
AMS Subject Classification: 54F99, 54D30, 54E35