On $\omega _1$ a Hausdorff inverse Lindelöf non Lindelöf topology has been constructed.
@article{118742,
author = {Viacheslav I. Malykhin},
title = {Distinguishing Lindel\"ofness and inverse Lindel\"ofness},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {36},
year = {1995},
pages = {171-176},
zbl = {0838.54004},
mrnumber = {1334424},
language = {en},
url = {http://dml.mathdoc.fr/item/118742}
}
Malykhin, Viacheslav I. Distinguishing Lindelöfness and inverse Lindelöfness. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 171-176. http://gdmltest.u-ga.fr/item/118742/
Inverse compactness, to appear. | MR 1320251 | Zbl 0920.54024
Inverse compactness versus compactness, to appear. | MR 1462390 | Zbl 0920.54024
An introduction to $\beta ømega $, in Handbook of Set-Theoretic Topology (K. Kunen and J.E. Vaughan, eds.), Elsevier Science Publishers B.V., 1984, p. 503. | MR 0776630 | Zbl 0555.54004