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http://hdl.handle.net/11320/7630
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Pole DC | Wartość | Język |
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dc.contributor.author | Koch, Sebastian | - |
dc.date.accessioned | 2019-03-04T10:34:24Z | - |
dc.date.available | 2019-03-04T10:34:24Z | - |
dc.date.issued | 2018/10/01 | - |
dc.identifier.citation | Formalized Mathematics, Volume 26, Issue 3, Pages 209-222 | - |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/7630 | - |
dc.description.abstract | This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4]. Let {Ti}i∈I be a family of topological spaces. The prebasis of the product space T =Qi∈ITi is defined in [5] as the set of all π−1i(V ) with i ∈ I and V open in Ti. Here it is shown that the basis generated by this prebasis consists exactly of the sets Qi∈IVi with Vi open in Ti and for all but finitely many i ∈ I holds Vi = Ti. Given I = {a} we have T ∼= Ta, given I = {a, b} with a 6= b we have T ∼= Ta × Tb. Given another family of topological spaces {Si}i∈I such that Si ∼= Ti for all i ∈ I, we have S = Qi∈ISi ∼= T . If instead Si is a subspace of Ti for each i ∈ I, then S is a subspace of T . These results are obvious for mathematicians, but formally proven here by means of the Mizar system [3], [2]. | - |
dc.language.iso | en | - |
dc.publisher | DeGruyter Open | - |
dc.subject | topology | - |
dc.subject | product spaces | - |
dc.title | Some Remarks about Product Spaces | - |
dc.type | Article | - |
dc.identifier.doi | 10.2478/forma-2018-0019 | - |
dc.description.Affiliation | Johannes Gutenberg University, Mainz, Germany | - |
dc.description.references | Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics,5(4):485–492, 1996. | - |
dc.description.references | 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. | - |
dc.description.references | Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. | - |
dc.description.references | Nicolas Bourbaki. Elements de Mathematique, volume Topologie Generale. HERMANN, troisieme edition, 1960. | - |
dc.description.references | Jarosław Gryko. Injective spaces. Formalized Mathematics,7(1):57–62, 1998. | - |
dc.description.references | John L. Kelley. General Topology, volume 27 of Graduate Texts in Mathematics. Springer-Verlag, 1955. | - |
dc.description.references | James Raymond Munkres. Topology. Prentice-Hall, Upper Saddle River, NJ, 2 edition, 2000. | - |
dc.description.references | Adam Naumowicz. On the characterization of collineations of the Segre product of strongly connected partial linear spaces. Formalized Mathematics, 13(1):125–131, 2005. | - |
dc.description.references | Bartłomiej Skorulski. The Tichonov Theorem. Formalized Mathematics, 9(2):373–376, 2001. | - |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 26 | - |
dc.description.issue | 3 | - |
dc.description.firstpage | 209 | - |
dc.description.lastpage | 222 | - |
dc.identifier.citation2 | Formalized Mathematics | - |
dc.identifier.orcid | 0000-0002-9628-177X | - |
Występuje w kolekcji(ach): | Formalized Mathematics, 2018, Volume 26, Issue 3 |
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