Lindelöf indestructibility, topological games and selection principles

By Marion Scheepers, Franklin D. Tall. 2009

51 citations recorded by the source corpus.

Arhangel’skii proved that if a first countable Hausdorff space is Lindelof, then its cardinality is at most 2א0 . Such a clean upper bound for Lindelof spaces in the larger class of spaces whose points are Gδ has been more elusive. In this paper we continue the agenda started in [51], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelof property. Infinite games and selection principles, especially the Rothberger property, are essential tools in our investigations . A topological space is Lindelof if each open cover contains a countable subset that covers the space. Alexandrov asked if in the class of first countable Hausdorff spaces, every Lindelof space has cardinality at most 2א0 . 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 Lindelof space has cardinality less than the least measurable cardinal. Juhasz [26] showed that this bound is sharp: There are such Lindelof spaces of arbitrarily large cardinality below the least measurable cardinal. Juhasz’s examples are not Hausdorff spaces and the cardinality of the underlying spaces has countable cofinality. Shelah ∗Research supported by Grant A-7354 of the Natural Sciences and Engineering Research Council of Canada 2010 Mathematics Subject Classification: 54A25, 54A35, 54D20

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/41721df18bb4d23938eb83292769e958e7dfbcd8