Measurable cardinals and the cardinality of Lindelöf spaces (Q972499)

A \textit{points \(G_{\delta}\) space} is one in which every singleton is a countable intersection of open sets. A space is \textit{indestructibly Lindelöf} iff it remains Lindelöf after forcing with a countably closed forcing notion. Arhangelskii proved that every points \(G_{\delta}\) Lindelöf space has size less than the least measurable cardinal. An old theorem of Tall says that if there is, consistently, a supercompact cardinal, then there is a model of set theory in which \(2^{\aleph_0} = \aleph_1\) and a point \(G_{\delta}\) indestructibly Lindelöf space has size at most \(\aleph_1\). This paper weakens the hypothesis of Tall's theorem to ``if there is, consistently, a measurable cardinal,'' and derives more diverse conclusions. For example: Theorem: If it is consistent that there is a measurable cardinal \(\kappa\) and \(\kappa > \aleph_{\alpha} \geq \aleph_0\) then there is a model in which \(2^{\aleph_0} = \aleph_{\alpha+1}\) and every point \(G_{\delta}\) indestructibly Lindelöf space has cardinality \(\leq 2^{\aleph_0}\). The technique uses the following variation of the weakly precipitous ideal game: Fix an ideal \(J\). There are two players. The length of the game is \(\omega\). In inning \(n\), player I chooses a set \(O_n\) in \(J^+\), II chooses \(T_n\) in \(J^+\), and \(O_{n+1} \subset T_n\). II wins iff \(\bigcap_{n < \omega}T_n\) is in \(J^+\); otherwise I wins. [In the weakly precipitous game, II wins iff \(\bigcap_{n < \omega}T_n \neq \emptyset\); if I has no winning strategy for this game, \(J\) is called weakly precipitous.] This game is then used to determine strategies for the following game: Fix a space \(X\). There are two players. The length of the game is \(\omega_1\). In inning \(\gamma\), player I chooses an open cover of \(X\), and player II chooses a set in the cover. II wins the came if the sets it picks covers \(X\); otherwise \(I\) wins. There is also a result on Rothberger subspaces of indestructibly Lindelöf spaces.