Abstract.In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by G δ subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space X, then X is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space X, then the G δ -topology on X is Lindelöf.

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Abstract. In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to ω 1 -sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba's "ℵ 1 -Borel Conjecture" is equiconsistent with the existence of an inaccessible cardinal. IntroductionThe question of whether the Lindelöf property of topological spaces is preserved by countably closed forcing is of interest in conjunction with A. V. Arhangel'skiȋ's classic problem of whether Lindelöf T 2 spaces with points G δ have cardinality not exceeding the continuum [1]. For a survey of this problem, see [33]. This was later improved by M. Scheepers [26], who replaced "supercompact" by "measurable". "Points G δ " was then improved to "pseudocharacter ≤ ℵ 1 " by Tall and T. Usuba in [35]. Scheepers and Tall [27] noticed that indestructibility of a Lindelöf space is equivalent to player One not having a winning strategy in an ω 1 -length generalization of the Rothberger game introduced by F. Galvin in [9]. Definition 1.3. Let X be a topological space and α be an ordinal. We denote by G α 1 (O X , O X ) the game defined as follows. In each inning ξ ∈ α, player One chooses an open cover U ξ of X, and then player Two picks V ξ ∈ U ξ . Two wins the play if X = {V ξ : ξ ∈ α}; otherwise, One is the winner. 2010 Mathematics Subject Classification. Primary 54D30; Secondary 03E55, 54D20, 54F05, 54G20, 91A44.

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