Baire spaces, $G_\delta$'s, and directed complete spaces
Keywords:
Baire space, alpha-favorable, Banach-Mazur game, Choquet game, Gdelta, subcompact, directed completeAbstract
We discuss the proof of the Baire Category Theorem. We review Cech-completeness, pseudocompleteness, and various forms of $\alpha$-favorability. We present recent results about subcompactness, directed completeness, the Banach-Mazur game, and the strong Choquet game. A recurring theme is whether a completeness property is inherited by $G_\delta$ subspaces. We prove that the class of regular directed complete spaces is exactly the class of $G_\delta$ subspaces of subcompact spaces.
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