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DISTINGUISHING PERFECT SET PROPERTIES IN SEPARABLE METRIZABLE SPACES

Published online by Cambridge University Press:  22 January 2016

ANDREA MEDINI
ANDREA MEDINI*
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA WÄHRINGER STRASSE 25 A-1090 WIEN, AUSTRIAE-mail: [email protected]: http://www.logic.univie.ac.at/∼medinia2/
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Abstract

All spaces are assumed to be separable and metrizable. Our main result is that the statement “For every space X, every closed subset of X has the perfect set property if and only if every analytic subset of X has the perfect set property” is equivalent to b > ω1 (hence, in particular, it is independent of ZFC). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement “For every space X, if every Γ subset of X has the perfect set property then every Γ′ subset of X has the perfect set property” as Γ, Γ′ range over all pointclasses of complexity at most analytic or coanalytic.

Along the way, we define and investigate a property of independent interest. We will say that a subset W of 2ω has the Grinzing property if it is uncountable and for every uncountable Y ⊆ W there exists an uncountable collection consisting of uncountable subsets of Y with pairwise disjoint closures in 2ω. The following theorems hold.

  1. (1) There exists a subset of 2ω with the Grinzing property.

  2. (2) Assume MA + ¬CH. Then 2ω has the Grinzing property.

  3. (3) Assume CH. Then 2ω does not have the Grinzing property.

The first result was obtained by Miller using a theorem of Todorčević, and is needed in the proof of our main result.

Keywords

Perfect set propertydropping Polishnessunbounding numberGrinzing property
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 166 - 180
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

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