Analytically heavy spaces: Analytic Cantor and Analytic Baire Theorems |
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Authors: | AJ Ostaszewski |
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Institution: | Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom |
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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. |
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Keywords: | 54H05 28A05 26A03 |
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