Geodesic
Convergence in compacta and linear Lindelöfness
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 1, pp. 159-166

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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.
Classification : 54A25, 54D30, 54E35, 54F99
Keywords: point of complete accumulation; linearly Lindelöf space; local compactness; first countability; $\kappa $-accessible diagonal 
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     author = {Arhangel'skii, A. V. and Buzyakova, R. Z.},
     title = {Convergence in compacta and linear {Lindel\"ofness}},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {159--166},
     publisher = {mathdoc},
     volume = {39},
     number = {1},
     year = {1998},
     mrnumber = {1623006},
     zbl = {0937.54022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998__39_1_a15/}
}
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Arhangel'skii, A. V.; Buzyakova, R. Z. Convergence in compacta and linear Lindelöfness. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 1, pp. 159-166. http://geodesic.mathdoc.fr/item/CMUC_1998__39_1_a15/