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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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Volume 24, Issue 3 (Sep 2016)

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Quasi-uniform Space

Roland Coghetto
  • Corresponding author
  • Rue de la Brasserie 5 7100 La Louvière, Belgium
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0017

Abstract

In this article, using mostly Pervin [9], Kunzi [6], [8], [7], Williams [11] and Bourbaki [3] works, we formalize in Mizar [2] the notions of quasiuniform space, semi-uniform space and locally uniform space.

We define the topology induced by a quasi-uniform space. Finally we formalize from the sets of the form ((X \ Ω) × X) ∪ (X × Ω), the Csaszar-Pervin quasi-uniform space induced by a topological space.

MSC: 54E15; 03B35

Keywords: quasi-uniform space; quasi-uniformity; Pervin space; Csaszar-Pervin quasi-uniformity

MML: identifier: UNIFORM2; version: 8.1.05 5.37.1275

References

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  • [2] Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: [Crossref] [Crossref]

  • [3] Nicolas Bourbaki. General Topology: Chapters 1-4. Springer Science and Business Media, 2013.

  • [4] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

  • [5] Roland Coghetto. Convergent filter bases. Formalized Mathematics, 23(3):189-203, 2015. doi: [Crossref] [Crossref]

  • [6] Hans-Peter A. Künzi. Quasi-uniform spaces - eleven years later. In Topology Proceedings, volume 18, pages 143-171, 1993.

  • [7] Hans-Peter A. Künzi. An introduction to quasi-uniform spaces. Beyond Topology, 486: 239-304, 2009.

  • [8] Hans-Peter A. Künzi and Carolina Ryser. The Bourbaki quasi-uniformity. In Topology Proceedings, volume 20, pages 161-183, 1995.

  • [9] William J. Pervin. Quasi-uniformization of topological spaces. Mathematische Annalen, 147(4):316-317, 1962.

  • [10] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.

  • [11] James Williams. Locally uniform spaces. Transactions of the American Mathematical Society, 168:435-469, 1972.

  • [12] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.

About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0017. Export Citation

© by Roland Coghetto. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License (CC BY-SA 3.0 LEGALCODE)

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