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Some examples of Borel-inseparable pairs of coanalylic sets

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  26 February 2010

Howard Becker
Howard Becker
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina, 29208, U.S.A.
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Extract

The purpose of this paper is to give some natural examples of Borel-inseparable pairs of coanalytic sets in Polish spaces.

A Polish space is a topological space homeomorphic to a separable complete metric space. In this paper, all spaces are uncountable Polish spaces. A pointset is analytic (or ) if it is the continuous image of a Borel set (in any space), or equivalently, the projection of a Borel set, and is coanalytic (or ) if it is the complement of an analytic set. The class of analytic sets is closed under countable unions and intersections, images and preimages by Borel measurable functions, and projections; it is not closed under complements, hence there is an analytic set which is not Borel.

MSC classification

Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 72 - 79
Copyright
Copyright © University College London 1986

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