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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytic groups and pushing small sets apart


Author: Jan van Mill
Journal: Trans. Amer. Math. Soc. 361 (2009), 5417-5434
MSC (2000): Primary 54H05, 54E52, 03E15
Published electronically: May 12, 2009
MathSciNet review: 2515817
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Abstract: We say that a space $ X$ has the separation property provided that if $ A$ and $ B$ are subsets of $ X$ with $ A$ countable and $ B$ first category, then there is a homeomorphism $ f\colon X\to X$ such that $ f(A)\cap B=\emptyset$. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space $ X$ come from the homeomorphisms given by an action of an analytic group, then $ X$ is Polish. Several examples are also presented.


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Additional Information

Jan van Mill
Affiliation: Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081$^{a}$, 1081 HV Amsterdam, The Netherlands
Email: vanmill@few.vu.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04665-0
PII: S 0002-9947(09)04665-0
Keywords: Analytic group, meager set, countable set, Polish space.
Received by editor(s): May 29, 2007
Received by editor(s) in revised form: October 18, 2007
Published electronically: May 12, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.