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THE COMPLEXITY OF HOMEOMORPHISM RELATIONS ON SOME CLASSES OF COMPACTA

Part of: Special properties Set theory Connections with other structures, applications

Published online by Cambridge University Press:  18 June 2020

PAWEŁ KRUPSKI  and
BENJAMIN VEJNAR
PAWEŁ KRUPSKI
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE FACULTY OF FUNDAMENTAL PROBLEMS OF TECHNOLOGY WROCłAW UNIVERSITY OF SCIENCE AND TECHNOLOGYWROCłAW, POLANDE-mail: pawel.krupski@pwr.edu.pl
BENJAMIN VEJNAR
Affiliation:
DEPARTMENT OF MATHEMATICAL ANALYSIS FACULTY OF MATHEMATICS AND PHYSICS CHARLES UNIVERSITYPRAGUE, CZECHIAE-mail: vejnar@karlin.mff.cuni.cz
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Abstract

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

Keywords

Borel reductioncountable structuresuniversal orbit equivalence relationabsolute retractLCn-compactumregular continuum

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 54F15: Continua and generalizations
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 733 - 748
Copyright
© The Association for Symbolic Logic 2020

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Footnotes

Dedicated to the memory of Věra Trnková.

References

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