Ostaszewski, Adam (2011) Analytically heavy spaces: analytic cantor and analytic Baire theorems. Topology and its applications, 158 (3). pp. 253-275. ISSN 0166-8641
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) , Haworth and McCoy (1977) ). 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.
|Additional Information:||© 2011 Elsevier B.V.|
|Uncontrolled Keywords:||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|
|Library of Congress subject classification:||Q Science > QA Mathematics|
|Sets:||Departments > Mathematics|
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