Lindelöf indestructibility, topological games and selection principles

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 20 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are Gδ has been more elusive. In this paper we continue the agenda started in [50], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property, are essential tools in our investigations. A topological space is Lindelöf if each open cover contains a countable subset that covers the space. Alexandrov asked if in the class of first countable Hausdorff spaces, every Lindelöf space has cardinality at most 20 . Arhangel’skii [1] proved that the answer is “yes”. This focuses attention on the larger class of spaces in which “points are Gδ” i.e., the class of topological spaces in which each point is an intersection of countably many open sets. Such spaces are T1 but not necessarily Hausdorff. Arhangel’skii showed that in the class of spaces with points Gδ each Lindelöf space has cardinality less than the least measurable cardinal. Juhász [26] showed that this bound is sharp: There are such Lindelöf spaces of arbitrary large cardinality below the least measurable cardinal. Juhász’s examples are not Hausdorff spaces and the cardinality of the underlying spaces has countable cofinality. Shelah [44] showed that no Lindelöf space with points Gδ can be of weakly compact cardinality. Gorelic [19] showed that it is relatively consistent that the Continuum Hypothesis (CH) holds, that 21 is arbitrarily large, and there is a zero-dimensional regular Lindelöf space with points Gδ and of cardinality 2 א1 . This improved earlier results of Shelah [44], Shelah-Stanley [45] and Velleman [58] which showed that either by countably closed forcing, or by assuming V=L, one could obtain such a space of cardinality א2, consistent with CH. Little else is known about cardinality Research supported by Grant A-7354 of the Natural Sciences and Engineering Research Council of Canada