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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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In This Section
Volume 24, Issue 3 (Sep 2016)

Issues

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-2016-0018

Abstract

In this article, we formalize in Mizar [1] the notion of uniform space introduced by André Weil using the concepts of entourages [2].

We present some results between uniform space and pseudo metric space. We introduce the concepts of left-uniformity and right-uniformity of a topological group.

Next, we define the concept of the partition topology. Following the Vlach’s works [11, 10], we define the semi-uniform space induced by a tolerance and the uniform space induced by an equivalence relation.

Finally, using mostly Gehrke, Grigorieff and Pin [4] works, a Pervin uniform space defined from the sets of the form ((X\A) × (X\A)) ∪ (A×A) is presented.

MSC: 54E15; 03B35

Keywords: uniform space; uniformity; pseudo-metric space; topological group; partition topology; Pervin uniform space

MML: identifier: UNIFORM3; version: 8.1.05 5.37.1275

References

  • [1] 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]

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

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

  • [4] Mai Gehrke, Serge Grigorieff, and Jean-Éric Pin. A topological approach to recognition. In Automata, Languages and Programming, pages 151-162. Springer, 2010.

  • [5] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.

  • [6] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.

  • [7] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.

  • [8] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

  • [9] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5): 855-864, 1990.

  • [10] Milan Vlach. Algebraic and topological aspects of rough set theory. In Fourth International Workshop on Computational Intelligence & Applications, IEEE SMC Hiroshima Chapter, Hiroshima University, Japan, December 10&11, 2008.

  • [11] Milan Vlach. Topologies of approximation spaces of rough set theory. In Interval/ Probabilistic Uncertainty and Non-Classical Logics, pages 176-186. Springer, 2008.

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-2016-0018. 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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