Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems

作者：Ostaszewski A J^*

来源：Topology and Its Applications, 2011, 158(3): 253-275.

DOI：10.1016/j.topol.2010.11.001

• Analytic
• kappa-analytic
• Analytically heavy
• Weakly alpha-favourable
• Heavy sets
• Irreducible submap
• Cantor Theorem
• Baire space
• Banach-Mazur games
• Choquet games
• Luzin separation
• Fine topology
• Density topology
• Gandy-Harrington topology
• Ellentuck topology
• O'Malley topologies
• Effros Theorem

Motivated by recent work, we establish the Baire Theorem in the broad context afforded by weak forms of completeness implied by analyticity and,kappa-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 alpha-favourability conditions (and therefrom variants of the Baire Theorem), in a form in which the principal role is played by kappa-analytic (in particular analytic) sets that are 'heavy' (everywhere large in the sense of some sigma-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 kappa-analytic form.