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Linear homeomorphisms of function spaces and the position of a space in its compactification

Part of: Linear function spaces and their duals Fairly general properties Maps and general types of spaces defined by maps Game theory Basic constructions

Published online by Cambridge University Press:  28 November 2023

Mikołaj Krupski Open the ORCID record for Mikołaj Krupski [Opens in a new window]
Mikołaj Krupski*
Affiliation:
Institute of Mathematics, University of Warsaw, Warszawa, Poland
*
e-mail: mkrupski@mimuw.edu.pl
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Abstract

An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

Keywords

Function spacepointwise convergence topologyMenger spaceHurewicz spacelinear homeomorphism

MSC classification

Primary: 46E10: Topological linear spaces of continuous, differentiable or analytic functions 54C35: Function spaces 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.) 54D40: Remainders
Secondary: 54C60: Set-valued maps 54B20: Hyperspaces 91A44: Games involving topology or set theory
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 22
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The author was partially supported by the NCN (National Science Centre, Poland) research Grant No. 2020/37/B/ST1/02613.

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