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