A simple proof of Tychonoff’s theorem (the compactness of the product of compact spaces) as a direct application of Zorn’s lemma is given. In contrast to the classical Cartan-Bourbaki proof which uses Zorn’s lemma twice, our proof uses it only once.
| Published in |
Pure and Applied Mathematics Journal (Volume 4, Issue 2-1)
This article belongs to the Special Issue Abridging over Troubled Water---Scientific Foundation of Engineering Subjects |
| DOI | 10.11648/j.pamj.s.2015040201.13 |
| Page(s) | 14-17 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Tychonoff’s Theorem, Compact Space, Product Space, Zorn’s Lemma, Axiom of Choice, Ultrafilter Theorem
| [1] | N. Bourbaki, General Topology: Chapters 1-4, Springer-Verlag 1989. |
| [2] | T. J. Jech, The Axiom of Choice,North-Holland, 1973. |
| [3] | J. L. Kelly, General Topology, D. Van Nostrand Company Inc. 1955. |
| [4] | G. H. Moore, Zermelo’s Axiom of Choice: Its Origins, Development, & Influence, Springer-Verlag 1982. |
| [5] | A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Ann. 102 (1929), 544-561. |
| [6] | S. Willard, General Topology, Addison-Wesley 1970. |
APA Style
Haruo Tsukada. (2014). Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma. Pure and Applied Mathematics Journal, 4(2-1), 14-17. https://doi.org/10.11648/j.pamj.s.2015040201.13
ACS Style
Haruo Tsukada. Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma. Pure Appl. Math. J. 2014, 4(2-1), 14-17. doi: 10.11648/j.pamj.s.2015040201.13
AMA Style
Haruo Tsukada. Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma. Pure Appl Math J. 2014;4(2-1):14-17. doi: 10.11648/j.pamj.s.2015040201.13
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Download@article{10.11648/j.pamj.s.2015040201.13,
author = {Haruo Tsukada},
title = {Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma},
journal = {Pure and Applied Mathematics Journal},
volume = {4},
number = {2-1},
pages = {14-17},
doi = {10.11648/j.pamj.s.2015040201.13},
url = {https://doi.org/10.11648/j.pamj.s.2015040201.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040201.13},
abstract = {A simple proof of Tychonoff’s theorem (the compactness of the product of compact spaces) as a direct application of Zorn’s lemma is given. In contrast to the classical Cartan-Bourbaki proof which uses Zorn’s lemma twice, our proof uses it only once.},
year = {2014}
}
TY - JOUR T1 - Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma AU - Haruo Tsukada Y1 - 2014/12/27 PY - 2014 N1 - https://doi.org/10.11648/j.pamj.s.2015040201.13 DO - 10.11648/j.pamj.s.2015040201.13 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 14 EP - 17 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.s.2015040201.13 AB - A simple proof of Tychonoff’s theorem (the compactness of the product of compact spaces) as a direct application of Zorn’s lemma is given. In contrast to the classical Cartan-Bourbaki proof which uses Zorn’s lemma twice, our proof uses it only once. VL - 4 IS - 2-1 ER -
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