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Transactions of the American Mathematical Society

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Embeddability on functions: Order and chaos


Authors: Raphaël Carroy, Yann Pequignot and Zoltán Vidnyánszky
Journal: Trans. Amer. Math. Soc. 371 (2019), 6711-6738
MSC (2010): Primary 03E15, 26A21, 54C05, 54C25; Secondary 06A07
DOI: https://doi.org/10.1090/tran/7739
Published electronically: January 16, 2019
MathSciNet review: 3937342
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the quasi-order of topological embeddability on definable functions between Polish 0-dimensional spaces.

We consider the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order.

We also investigate the existence of maximal elements with respect to embeddability in a given Baire class.We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.


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Additional Information

Raphaël Carroy
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien, Austria
Email: raphael.carroy@univie.ac.at

Yann Pequignot
Affiliation: Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montréal, Québec H3A 0B9, Canada
Email: yann.pequignot@mcgill.ca

Zoltán Vidnyánszky
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien, Austria
Email: zoltan.vidnyanszky@univie.ac.at

DOI: https://doi.org/10.1090/tran/7739
Received by editor(s): February 23, 2018
Received by editor(s) in revised form: September 16, 2018, and October 8, 2018
Published electronically: January 16, 2019
Additional Notes: The first author was supported by FWF Grant P28153
The second author gratefully acknowledges the support of the Swiss National Science Foundation (SNF) through Grant P2LAP2$_$164904.
The third author was supported by FWF Grant P29999. He was also partially supported by the National Research, Development and Innovation Office (NKFIH), Grants 113047 and 104178.
Article copyright: © Copyright 2019 American Mathematical Society