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Borel and projective sets from the point of view of compact sets

Published online by Cambridge University Press:  24 October 2008

Kenneth Kunen
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Arnold W. Miller
Affiliation:
Department of Mathematics, The University of Texas, Austin, Texas 78712

Extract

In this paper we prove several results concerning the complexity of a set relative to compact sets. We prove that for any Polish space X and Borel set BX, if B is not , then there exists a compact zero-dimensional PX such that pX is not . We also show that it is consistent with ZFC that, for any A ⊆ ωω, if for all compact K ⊆ ωωAK is , then A is . This generalizes to in place of assuming the consistency of some hypotheses involving determinacy. We give an alternative proof of the following theorem of Saint-Raymond. Suppose X and Y are compact metric spaces and f is a continuous surjection of X onto Y. Then, for any AY, A is in Y iff f−1(A) is in X. The non-trivial part of this result is to show that taking pre-images cannot reduce the Borel complexity of a set. The techniques we use are the definability of forcing and Wadge games.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1983

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