We prove that every Tychonoff strongly monotonically monolithic star countable space is Lindelöf, which solves a question posed by O.T. Alas et al. We also use this result to generalize a metrization theorem for strongly monotonically monolithic spaces. At the end of this paper, we study the extent of star countable spaces with k-in-countable bases, k ∈ ℤ.
@article{bwmeta1.element.doi-10_2478_s11533-012-0030-x,
author = {Zuoming Yu},
title = {A note on the extent of two subclasses of star countable spaces},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1067-1070},
zbl = {1243.54042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0030-x}
}
Zuoming Yu. A note on the extent of two subclasses of star countable spaces. Open Mathematics, Tome 10 (2012) pp. 1067-1070. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0030-x/
[1] Alas O.T., Junqueira L.R., van Mill J., Tkachuk V.V., Wilson R.G., On the extent of star countable spaces, Cent. Eur. J. Math., 2011, 9(3), 603–615 http://dx.doi.org/10.2478/s11533-011-0018-y | Zbl 1246.54017
[2] van Douwen E.K., Reed G.M., Roscoe, A.W., Tree I.J., Star covering properties, Topology Appl., 1991, 39(1), 71–103 http://dx.doi.org/10.1016/0166-8641(91)90077-Y | Zbl 0743.54007
[3] Gruenhage G., Infinite games and generalizations of first-countable spaces, General Topology and Appl., 1976, 6(3), 339–352 http://dx.doi.org/10.1016/0016-660X(76)90024-6
[4] Gruenhage G., The story of a topological game, Rocky Mountain J. Math., 2006, 36(6), 1885–1914 http://dx.doi.org/10.1216/rmjm/1181069351 | Zbl 1141.54020
[5] Heath R.W., Lindgren W.F., Weakly uniform bases, Houston J. Math., 1976, 2(1), 85–90 | Zbl 0318.54032
[6] van Mill J., Tkachuk V.V., Wilson R.G., Classes defined by stars and neighbourhood assignments, Topology Appl., 2007, 154(10), 2127–2134 http://dx.doi.org/10.1016/j.topol.2006.03.029 | Zbl 1131.54022
[7] Tkachuk V.V., Monolithic spaces and D-spaces revisited, Topology Appl., 2009, 156(4), 840–846 http://dx.doi.org/10.1016/j.topol.2008.11.001 | Zbl 1165.54009