Volume 4, Issue 2-1, March 2015, Page: 14-17
Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma
Haruo Tsukada, Department of Information and Computer Science, Faculty of Humanity-Oriented Science and Engineering, Kindai University, 11-6 Kayanomori, Iizuka, Fukuoka, 820-8555, Japan
Received: Nov. 25, 2014;       Accepted: Dec. 11, 2014;       Published: Dec. 27, 2014
DOI: 10.11648/j.pamj.s.2015040201.13      View  2664      Downloads  172
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.
Keywords
Tychonoff’s Theorem, Compact Space, Product Space, Zorn’s Lemma, Axiom of Choice, Ultrafilter Theorem
To cite this article
Haruo Tsukada, Tychonoff’s Theorem as a Direct Application of Zorn’s Lemma, Pure and Applied Mathematics Journal. Special Issue: Abridging over Troubled Water---Scientific Foundation of Engineering Subjects. Vol. 4, No. 2-1, 2015, pp. 14-17. doi: 10.11648/j.pamj.s.2015040201.13
Reference
[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.
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