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Between Polish and completely Baire

Between Polish and completely Baire All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. (1) X is Polish. (2) For every countable crowded $${Q \subseteq X}$$ Q ⊆ X there exists a crowded $${Q'\subseteq Q}$$ Q ′ ⊆ Q with compact closure. (3) Every closed subspace of X is either scattered or it contains a homeomorphic copy of $${2^\omega}$$ 2 ω . (4) Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications $${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}$$ ( 1 ) → ( 2 ) → ( 3 ) → ( 4 ) hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a $${\sf ZFC}$$ ZFC counterexample and a consistent definable counterexample of lowest possible complexity to the implication $${(i) \leftarrow (i + 1)}$$ ( i ) ← ( i + 1 ) for $${i = 1, 2, 3}$$ i = 1 , 2 , 3 . For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Between Polish and completely Baire

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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-014-0409-4
Publisher site
See Article on Publisher Site

Abstract

All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. (1) X is Polish. (2) For every countable crowded $${Q \subseteq X}$$ Q ⊆ X there exists a crowded $${Q'\subseteq Q}$$ Q ′ ⊆ Q with compact closure. (3) Every closed subspace of X is either scattered or it contains a homeomorphic copy of $${2^\omega}$$ 2 ω . (4) Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications $${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}$$ ( 1 ) → ( 2 ) → ( 3 ) → ( 4 ) hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a $${\sf ZFC}$$ ZFC counterexample and a consistent definable counterexample of lowest possible complexity to the implication $${(i) \leftarrow (i + 1)}$$ ( i ) ← ( i + 1 ) for $${i = 1, 2, 3}$$ i = 1 , 2 , 3 . For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Oct 30, 2014

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