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CONTINUITY OF UNIVERSALLY MEASURABLE HOMOMORPHISMS

Part of: Abstract harmonic analysis Set theory Topological and differentiable algebraic systems

Published online by Cambridge University Press:  13 August 2019

CHRISTIAN ROSENDAL
CHRISTIAN ROSENDAL*
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA; rosendal.math@gmail.com

Abstract

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Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo $\text{ZF}+\text{DC}$, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on $\{0,1\}^{\mathbb{N}}$ has finite chromatic number.

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 22A05: Structure of general topological groups 43A05: Measures on groups and semigroups, etc.
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e5
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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