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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


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Characterizing realcompact locales via remainders

Themba Dube
  • Corresponding author
  • Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa
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Published Online: 2019-05-16 | DOI: https://doi.org/10.1515/gmj-2019-2027

Abstract

We prove that a completely regular locale L is realcompact if and only if the “remainder” βLL is the join of the zero-sublocales of βL that miss L. This extends a result of Mrówka which characterizes realcompact spaces in terms of their remainders in Stone–Čech compactifications. We prove that βLL is Lindelöf if and only if L is of countable type, where the latter is defined for locales exactly as for spaces, subject to replacing subspaces with sublocales.

Keywords: Locale; sublocale; realcompact; Lindelöf; remainder; countable type

MSC 2010: 06D22; 54B05; 54D20; 54D60

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About the article

Received: 2017-07-15

Revised: 2018-02-11

Accepted: 2018-02-16

Published Online: 2019-05-16


The author acknowledges funding from the National Research Foundation of South Africa through a research grant with Grant Number 93514.


Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2027.

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