Products, the Baire category theorem, and the axiom of dependent choice

Horst Herrlich; Kyriakos Keremedis

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 4, page 771-775
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Herrlich, Horst, and Keremedis, Kyriakos. "Products, the Baire category theorem, and the axiom of dependent choice." Commentationes Mathematicae Universitatis Carolinae 40.4 (1999): 771-775. <http://eudml.org/doc/248409>.

@article{Herrlich1999,
abstract = {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.},
author = {Herrlich, Horst, Keremedis, Kyriakos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of dependent choice; Baire category theorem; Baire space; (countably) compact; pseudocompact; Čech-complete; regular-closed; pseudo-complete; product spaces; axiom of choice; Baire category theorem; product},
language = {eng},
number = {4},
pages = {771-775},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Products, the Baire category theorem, and the axiom of dependent choice},
url = {http://eudml.org/doc/248409},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Herrlich, Horst
AU - Keremedis, Kyriakos
TI - Products, the Baire category theorem, and the axiom of dependent choice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 4
SP - 771
EP - 775
AB - 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.
LA - eng
KW - axiom of dependent choice; Baire category theorem; Baire space; (countably) compact; pseudocompact; Čech-complete; regular-closed; pseudo-complete; product spaces; axiom of choice; Baire category theorem; product
UR - http://eudml.org/doc/248409
ER -

References

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  2. Bourbaki N., Topologie générale, ch. IX., Paris, 1948. Zbl1107.54001
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  7. Fossy J., Morillon M., The Baire category property and some notions of compactness, preprint, 1995. Zbl0922.03070MR1624737
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  9. Hausdorff F., Grundzüge der Mengenlehre, Leipzig, 1914. Zbl1010.01031
  10. Herrlich H., T ν -Abgeschlossenheit und T ν -Minimalität, Math. Z. 88 (1965), 285-294. (1965) Zbl0139.40203MR0184191
  11. Herrlich H., Keremedis K., The Baire Category Theorem and Choice, to appear in Topology Appl. Zbl0991.54036MR1787859
  12. Howard P., Rubin J.E., Consequences of the axiom of choice, AMS Math. Surveys and Monographs 59, 1998. Zbl0947.03001MR1637107
  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. (1924) 
  14. Oxtoby J.C., Cartesian products of Baire spaces, Fund. Math. 49 (1961), 157-166. (1961) Zbl0113.16402MR0140638
  15. Pettey D.H., Products of regular-closed spaces, Topology Appl. 14 (1982), 189-199. (1982) Zbl0499.54017MR0667666

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