On the non-existence of certain group topologies

Christian Rosendal

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CONTENTS1. Introduction.................................................................................. 32. Baire spaces................................................................................ 53. The basic theorem..................................................................... 94. Cardinality properties; invariance of weight........................... 165. Classification of absolute Borel sets..................................... 226. Characterizations..........................................................................

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Anna Kucia (1990)

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R. A. Johnson showed that there is no translation-invariant Borel lifting for the measure algebra of ℝ/ℤ equipped with Haar measure, a result which was generalized by M. Talagrand to non-discrete locally compact abelian groups and by J. Kupka and K. Prikry to arbitrary non-discrete locally compact groups. In this paper we study analogs of these results for category algebras (the Borel σ-algebra modulo the ideal of first category sets) of topological groups. Our main results are for the...

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Turning Borel sets into clopen sets effectively

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We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be chosen in a hyperarithmetical way and using this we obtain some uniformity results.