Countable compact Hausdorff spaces need not be metrizable in ZF

Keremedis, Kyriakos; Tachtsis, Eleftherios

doi:10.1090/S0002-9939-06-08572-8

Countable compact Hausdorff spaces need not be metrizable in ZF
HTML articles powered by AMS MathViewer

by Kyriakos Keremedis and Eleftherios Tachtsis

Proc. Amer. Math. Soc. 135 (2007), 1205-1211

DOI: https://doi.org/10.1090/S0002-9939-06-08572-8

Published electronically: October 4, 2006

PDF | Request permission

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

Eric K. van Douwen, Horrors of topology without $\textrm {AC}$: a nonnormal orderable space, Proc. Amer. Math. Soc. 95 (1985), no. 1, 101–105. MR 796455, DOI 10.1090/S0002-9939-1985-0796455-5
S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 56 (1964/65), 325–345. MR 176925, DOI 10.4064/fm-56-3-325-345
C. Good and I. J. Tree, Continuing horrors of topology without choice, Topology Appl. 63 (1995), no. 1, 79–90. MR 1328621, DOI 10.1016/0166-8641(95)90010-1
C. Good, I. J. Tree, and W. S. Watson, On Stone’s theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1211–1218. MR 1425122, DOI 10.1090/S0002-9939-98-04163-X
H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics, Springer-Verlag Vol. 1876, Berlin, Heidelberg, 2006.
Paul Howard, Kyriakos Keremedis, Herman Rubin, and Jean E. Rubin, Disjoint unions of topological spaces and choice, MLQ Math. Log. Q. 44 (1998), no. 4, 493–508. MR 1654348, DOI 10.1002/malq.19980440408
Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. With 1 IBM-PC floppy disk (3.5 inch; WD). MR 1637107, DOI 10.1090/surv/059
Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0396271
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
J. L. Kelley, The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37 (1950), 75–76. MR 39982, DOI 10.4064/fm-37-1-75-76
Kyriakos Keremedis, Disasters in topology without the axiom of choice, Arch. Math. Logic 40 (2001), no. 8, 569–580. MR 1867681, DOI 10.1007/s001530100094
Kyriakos Keremedis and Eleftherios Tachtsis, Countable sums and products of metrizable spaces in ZF, MLQ Math. Log. Q. 51 (2005), no. 1, 95–103. MR 2099390, DOI 10.1002/malq.200410011
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
L. E. Ward Jr., A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc. 13 (1962), 757–758. MR 186537, DOI 10.1090/S0002-9939-1962-0186537-8
Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
E. S. Wolk, On theorems of Tychonoff, Alexander, and R. Rado, Proc. Amer. Math. Soc. 18 (1967), 113–115. MR 203680, DOI 10.1090/S0002-9939-1967-0203680-X