Measurable Cardinals and the Cardinality of Lindelöf Spaces

We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets. Call a topological space in which each singleton is a Gδ set a points Gδ space. A.V. Arhangel’skii proved that any points Gδ Lindelöf space must have cardinality less than the least measurable cardinal and asked whether for T2 spaces this cardinality upper bound could be improved. I. Juhasz constructed examples showing that for T1 spaces this upper bound is sharp. F.D. Tall, investigating Arhangel’skii’s problem, defined the class of indestructibly Lindelöf spaces. A Lindelöf space is indestructible if it remains Lindelöf after forcing with a countably closed forcing notion. He proved: Theorem 1 (F.D. Tall [15]). If it is consistent that there is a supercompact cardinal, then it is consistent that 20 = א1, and every points Gδ indestructibly Lindelöf space has cardinality ≤ א1. In this paper we show that the hypothesis that there is a supercompact cardinal can be weakened to the hypothesis that there exists a measurable cardinal. Our technique permits flexibility on the cardinality of the continuum. In Section 1 we review relevant information about ideals and the weakly precipitous ideal game. The relevance of the weakly precipitous ideal game to points Gδ spaces is given in Lemma 2. In Section 2 we consider the indestructibly Lindelöf spaces. A variation of the weakly precipitous ideal game is introduced. This variation is featured in the main result, Theorem 4: a cardinality restriction is imposed on the indestructibly Lindelöf spaces with points Gδ. In Section 3 we give the consistency strength of the hypothesis used in Theorem 4 and point out that mere existence of a precipitous ideal is insufficient to derive a cardinality bound on the indestructibly Lindelöf points Gδ spaces. In Section 4 we describe models of set theory in which the Continuum Hypothesis fails while there is a “small” upper bound on the cardinality of points Gδ indestructibly Lindelöf spaces. The notion of a Rothberger space appears in the paper. A spaceX is a Rothberger space if for each ω-sequence of open covers of X there is a sequence of open sets, then n-th belonging to the n-th cover, such that the terms of the latter sequence is an open cover of X . Rothberger spaces are indestructibly Lindelöf (but not conversely). More details about Rothberger spaces in this context can be found in [13]. Date: Version of September 18.