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Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma

Received: 25 November 2014     Accepted: 11 December 2014     Published: 27 December 2014
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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.

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), 2014. Published by Science Publishing Group

Keywords

Tychonoff’s Theorem, Compact Space, Product Space, Zorn’s Lemma, Axiom of Choice, Ultrafilter Theorem

References
[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.
Cite This Article
  • 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

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    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

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    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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  • @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}
    }
    

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    Y1  - 2014/12/27
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    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.
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Author Information
  • Department of Information and Computer Science, Faculty of Humanity-Oriented Science and Engineering, Kindai University, 11-6 Kayanomori, Iizuka, Fukuoka, 820-8555, Japan

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