We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product , where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms). This settles the open problem in Tachtsis (2012) on the relation of Q(n), n = 2,3,4,5, to .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm961-1-2016,
author = {Paul Howard and Eleftherios Tachtsis},
title = {On the set-theoretic strength of the n-compactness of generalized Cantor cubes},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {241-252},
zbl = {06602792},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm961-1-2016}
}
Paul Howard; Eleftherios Tachtsis. On the set-theoretic strength of the n-compactness of generalized Cantor cubes. Fundamenta Mathematicae, Tome 233 (2016) pp. 241-252. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm961-1-2016/