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  • 1 Instituto de Matemática, Universidade Federal da Bahia, Campus de Ondina, Av. Adhemar de Barros, S/N, Ondina, CEP 40170-110 Salvador, BA, Brazil
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Abstract

With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: “If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed”. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set.

  • [1] Bourbaki, N. 1939 Theorie des ensembles Éléments de mathématique Hermann Paris.

  • [2] Herrlich, H. 2006 Axiom of Choice Lecture Notes in Mathematics 1876 Springer-Verlag Berlin xiv + 194 pp.

  • [3] Howard, P., Rubin, J. E. 1998 Consequences of the Axiom of Choice Mathematical Surveys and Monographs 59 American Mathematical Society Providence RI viii + 432 pp.

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  • [4] de Jesus, J. P. C., Espacos métricos e topológicos na ausência do Axioma da Escolha (in Portuguese), MSc Dissertation, Federal University of Bahia (Salvador, Bahia, Brazil, 2010), 116 pp.

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  • [5] Schecther, E. 1992 Two topological equivalents of the Axiom of Choice Zeit. für Math. Logik und Grund. Math. 38 555557 .

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