Disasters in metric topology without choice

Eleftherios Tachtsis

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 165-174
  • ISSN: 0010-2628

Abstract

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We show that it is consistent with ZF that there is a dense-in-itself compact metric space ( X , d ) which has the countable chain condition (ccc), but X is neither separable nor second countable. It is also shown that X has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply the disjoint union of metrizable spaces is normal.

How to cite

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Tachtsis, Eleftherios. "Disasters in metric topology without choice." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 165-174. <http://eudml.org/doc/249011>.

@article{Tachtsis2002,
abstract = {We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X,d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply the disjoint union of metrizable spaces is normal.},
author = {Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of Choice; Axiom of Multiple Choice; Principle of Dependent Choice; Ordering Principle; metric spaces; separable metric spaces; second countable metric spaces; paracompact spaces; compact T$_2$ spaces; ccc spaces; axiom of choice; axiom of multiple choice; principle of dependent choice; ordering principle; metric spaces},
language = {eng},
number = {1},
pages = {165-174},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Disasters in metric topology without choice},
url = {http://eudml.org/doc/249011},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Tachtsis, Eleftherios
TI - Disasters in metric topology without choice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 165
EP - 174
AB - We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X,d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply the disjoint union of metrizable spaces is normal.
LA - eng
KW - Axiom of Choice; Axiom of Multiple Choice; Principle of Dependent Choice; Ordering Principle; metric spaces; separable metric spaces; second countable metric spaces; paracompact spaces; compact T$_2$ spaces; ccc spaces; axiom of choice; axiom of multiple choice; principle of dependent choice; ordering principle; metric spaces
UR - http://eudml.org/doc/249011
ER -

References

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  1. Cohen P.J., Set Theory and the Continuum Hypothesis, Benjamin, 1966. Zbl0182.01401MR0232676
  2. van Douwen E.K., Horrors of topology without AC: a non normal orderable space, Proc. Amer. Math. Soc. 95 (1985), 101-105. (1985) MR0796455
  3. Good C., Tree I.J., Continuing horrors of topology without choice, Topology Appl. 63 (1995), 79-90. (1995) Zbl0822.54001MR1328621
  4. Good C., Tree I.J., Watson W.S., On Stone's theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), 1211-1218. (1998) Zbl0893.54016MR1425122
  5. Herrlich H., Steprāns J., Maximal filters, continuity and choice principles, Quaestiones Math. 20 (1997), 697-705. (1997) MR1625478
  6. Herrlich H., Strecker G.E., When is ℕ Lindelöf?, Comment. Math. Univ. Carolinae 38.3 (1997), 553-556. (1997) Zbl0938.54008MR1485075
  7. Howard P., Keremedis K., Rubin H., Rubin J.E., Disjoint unions of topological spaces and choice, Math. Logic Quart. 44 (1998), 493-508. (1998) Zbl0922.03069MR1654348
  8. Howard P., Keremedis K., Rubin J.E., Stanley A., Paracompactness of metric spaces and the axiom of multiple choice, Math. Logic Quart. 46 (2000). (2000) Zbl0993.03059MR1755811
  9. Howard P., Keremedis K., Rubin J.E., Stanley A., Tachtsis E., Non-constructive properties of the real numbers, Math. Logic Quart. 47 (2001), 423-431. (2001) MR1847458
  10. Howard P., Rubin J.E., Consequences of the Axiom of Choice, Math. Surveys and Monographs 59, Amer. Math. Soc., Providence R.I., 1998. Zbl0947.03001MR1637107
  11. Jech T., The Axiom of Choice, North-Holland, Amsterdam, 1973. Zbl0259.02052MR0396271
  12. Keremedis K., Disasters in topology without the axiom of choice, Arch. Math. Logic, to appear. Zbl1027.03040MR1867681
  13. Keremedis K., Countable disjoint unions in topology and some weak forms of the axiom of choice, Arch. Math. Logic, submitted. 
  14. Keremedis K., Tachtsis E., Compact metric spaces and weak forms of the axiom of choice, Math. Logic Quart. 47 (2001), 117-128. (2001) Zbl0968.03057MR1808950
  15. Keremedis K., Tachtsis E., On Lindelöf metric spaces and weak forms of the axiom of choice, Math. Logic Quart. 46 (2000), 35-44. (2000) Zbl0952.03060MR1736648
  16. Kunen K., Set Theory, An Introduction to Independence Proofs, North-Holland, Amsterdam, 1983. Zbl0534.03026MR0756630
  17. Willard S., General Topology, Addison-Wesley Publ. Co., Reading, MA, 1968. Zbl1052.54001MR2048350

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