Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Countable compact Hausdorff spaces need not be metrizable in ZF


Authors: Kyriakos Keremedis and Eleftherios Tachtsis
Journal: Proc. Amer. Math. Soc. 135 (2007), 1205-1211
MSC (2000): Primary 03E25, 03E35, 54A35, 54D10, 54D30, 54D35, 54D65, 54D70, 54E35.
DOI: https://doi.org/10.1090/S0002-9939-06-08572-8
Published electronically: October 4, 2006
MathSciNet review: 2262927
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the existence of a countable, first countable, zero-dimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF.


References [Enhancements On Off] (What's this?)

  • 1. E. K. van Douwen, Horrors of topology without AC: A nonnormal orderable space, Proc. Amer. Math. Soc. 95 (1985), 101-105. MR 0796455 (87d:03130)
  • 2. S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 56 (1965), 325-345. MR 0176925 (31:1193)
  • 3. C. Good and I. Tree, Continuing horrors of topology without choice, Topology and its Applications 63 (1995), 79-90. MR 1328621 (96f:54003)
  • 4. C. Good, I. Tree, and S. Watson, On Stone's theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), 1211-1218. MR 1425122 (98f:03044)
  • 5. H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics, Springer-Verlag Vol. 1876, Berlin, Heidelberg, 2006.
  • 6. P. Howard, K. Keremedis, H. Rubin, and J. E. Rubin, Disjoint unions of topological spaces and choice, Math. Log. Quart. 44 (1998), 493-508. MR 1654348 (2000b:03179)
  • 7. P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Amer. Math. Soc., Math. Surveys and Monographs Vol. 59, Providence, RI, 1998. MR 1637107 (99h:03026)
  • 8. T. Jech, The Axiom of Choice, North-Holland Publ. Co., Amsterdam, 1973. MR 0396271 (53:139)
  • 9. T. Jech, Set theory, Academic Press, New York, 1978. MR 0506523 (80a:03062)
  • 10. J. L. Kelley, The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37 (1950), 75-76. MR 0039982 (12:626d)
  • 11. K. Keremedis, Disasters in topology without the axiom of choice, Arch. Math. Logic 40 (2001), 569-580. MR 1867681 (2002m:54005)
  • 12. K. Keremedis and E. Tachtsis, Countable Sums and Products of Metrizable Spaces in ZF, Math. Log. Quart. 51 (2005), 95-103. MR 2099390
  • 13. K. Kunen, Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam, 1983. MR 0756630 (85e:03003)
  • 14. L. E. Ward, A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc. 13 (1962), 757-758. MR 0186537 (32:3996)
  • 15. S. Willard, General Topology, Addison-Wesley Publ. Co. Boston, 1970. MR 0264581 (41:9173)
  • 16. E. S. Wolk, On theorems of Tychonoff, Alexander and R. Rado, Proc. Amer. Math. Soc. 18 (1967), 113-115. MR 0203680 (34:3529)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E25, 03E35, 54A35, 54D10, 54D30, 54D35, 54D65, 54D70, 54E35.

Retrieve articles in all journals with MSC (2000): 03E25, 03E35, 54A35, 54D10, 54D30, 54D35, 54D65, 54D70, 54E35.


Additional Information

Kyriakos Keremedis
Affiliation: Department of Mathematics, University of the Aegean, Karlovassi, 83 200, Samos, Greece
Email: kker@aegean.gr

Eleftherios Tachtsis
Affiliation: Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, 83 200, Samos, Greece
Email: ltah@aegean.gr

DOI: https://doi.org/10.1090/S0002-9939-06-08572-8
Keywords: Axiom of choice, weak axioms of choice, compact spaces, Hausdorff spaces, countable spaces, first countable spaces, second countable spaces, metrizable spaces, zero-dimensional spaces, paracompact spaces, metacompact spaces.
Received by editor(s): March 18, 2005
Received by editor(s) in revised form: May 15, 2005, November 5, 2005, and November 10, 2005
Published electronically: October 4, 2006
Communicated by: Julia Knight
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society