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Keywords:
axiom of dependent choice; Baire category theorem; Baire space; (countably) compact; pseudocompact; Čech-complete; regular-closed; pseudo-complete; product spaces
axiom of dependent choice; Baire category theorem; Baire space; (countably) compact; pseudocompact; Čech-complete; regular-closed; pseudo-complete; product spaces
Summary:
In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (i) The axiom of dependent choice. (ii) Products of compact Hausdorff spaces are Baire. (iii) Products of pseudocompact spaces are Baire. (iv) Products of countably compact, regular spaces are Baire. (v) Products of regular-closed spaces are Baire. (vi) Products of Čech-complete spaces are Baire. (vii) Products of pseudo-complete spaces are Baire.
References:
[1] Blaire C.E.: The Baire Category Theorem implies the principle of dependent choice. Bull. Acad. Math. Astronom. Phys. 25 (1977), 933-934. MR 0469765
[2] Bourbaki N.: Topologie générale. ch. IX., Paris, 1948. Zbl 1107.54001
[3] Brunner N.: Kategoriesätze und Multiples Auswahlaxiom. Zeitschr. f. Math. Logik und Grundlagen d. Math. 29 (1983), 435-443. MR 0716858 | Zbl 0526.03031
[4] Čech E.: On bicompact spaces. Ann. Math. 38 (1937), 823-844. MR 1503374
[5] Colmez J.: Sur les espaces précompacts. C.R. Acad. Paris 234 (1952), 1019-1021. MR 0055658 | Zbl 0047.16202
[6] Engelking R.: General Topology. Heldermann Verlag, 1989. MR 1039321 | Zbl 0684.54001
[7] Fossy J., Morillon M.: The Baire category property and some notions of compactness. preprint, 1995. MR 1624737 | Zbl 0922.03070
[8] Goldblatt R.: On the role of the Baire Category Theorem and Dependent Choice in the foundations of logic. J. Symbolic Logic 50 (1985), 412-422. MR 0793122 | Zbl 0567.03023
[9] Hausdorff F.: Grundzüge der Mengenlehre. Leipzig, 1914. Zbl 1010.01031
[10] Herrlich H.: $T_\nu$-Abgeschlossenheit und $T_\nu$-Minimalität. Math. Z. 88 (1965), 285-294. MR 0184191 | Zbl 0139.40203
[11] Herrlich H., Keremedis K.: The Baire Category Theorem and Choice. to appear in Topology Appl. MR 1787859 | Zbl 0991.54036
[12] Howard P., Rubin J.E.: Consequences of the axiom of choice. AMS Math. Surveys and Monographs 59, 1998. MR 1637107 | Zbl 0947.03001
[13] Moore R.L.: An extension of the theorem that no countable point set is perfect. Proc. Nat. Acad. Sci. USA 10 (1924), 168-170.
[14] Oxtoby J.C.: Cartesian products of Baire spaces. Fund. Math. 49 (1961), 157-166. MR 0140638 | Zbl 0113.16402
[15] Pettey D.H.: Products of regular-closed spaces. Topology Appl. 14 (1982), 189-199. MR 0667666 | Zbl 0499.54017
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