Topology from Neighbourhoods

• Autorzy: Roland Coghetto
• Języki publikacji: EN

Abstrakty

EN

Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology. We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].

Słowa kluczowe

EN

filter; topological space; neighbourhoods system

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 54A05: Topological spaces and generalizations (closure spaces, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2015
• Tom: 23
• Numer: 4
• Strony: 289-296

Daty

wydano

2015-12-01

otrzymano

2015-08-14

online

2016-03-25

Twórcy

autor

Roland Coghetto

• Rue de la Brasserie 5, 7100 La Louvière, Belgium

Bibliografia

• [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.
• [2] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.
• [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
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• [8] Grzegorz Bancerek, Noboru Endou, and Yuji Sakai. On the characterizations of compactness. Formalized Mathematics, 9(4):733–738, 2001.
• [9] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, 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:10.1007/978-3-319-20615-8 17.[Crossref]
• [10] Nicolas Bourbaki. General Topology: Chapters 1–4. Springer Science and Business Media, 2013.
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• [26] Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213–225, 1997.
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• [30] Josef Urban. Basic facts about inaccessible and measurable cardinals. Formalized Mathematics, 9(2):323–329, 2001.
• [31] Claude Wagschal. Topologie et analyse fonctionnelle. Hermann, 1995.
• [32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.
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Identyfikatory

DOI

10.1515/forma-2015-0023

bwmeta1.id-class.MML

FINTOPO7