Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

Abstract

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and K -analyticity, thereby adding to the ‘Baire space recognition literature’ (cf. Aarts and Lutzer (1974) [1], Haworth and McCoy (1977) [43]). We extend a metric result of van Mill, obtaining a generalization of Oxtoby's weak α-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by K -analytic (in particular analytic) sets that are ‘heavy’ (everywhere large in the sense of some σ-ideal). From this perspective fine-topology versions are derived, allowing a unified view of the Baire Theorem which embraces classical as well as generalized Gandy–Harrington topologies (including the Ellentuck topology), and also various separation theorems. A multiple-target form of the Choquet Banach–Mazur game is a primary tool, the key to which is a restatement of the Cantor Theorem, again in K -analytic form.