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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.
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 {\it the disjoint union of metrizable spaces is normal\/}.
[1] Cohen P.J.: Set Theory and the Continuum Hypothesis. Benjamin, 1966. MR 0232676 | Zbl 0182.01401
[2] van Douwen E.K.: Horrors of topology without AC: a non normal orderable space. Proc. Amer. Math. Soc. 95 (1985), 101-105. MR 0796455
[3] Good C., Tree I.J.: Continuing horrors of topology without choice. Topology Appl. 63 (1995), 79-90. MR 1328621 | Zbl 0822.54001
[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. MR 1425122 | Zbl 0893.54016
[5] Herrlich H., Steprāns J.: Maximal filters, continuity and choice principles. Quaestiones Math. 20 (1997), 697-705. MR 1625478
[6] Herrlich H., Strecker G.E.: When is $\Bbb N$ Lindelöf?. Comment. Math. Univ. Carolinae 38.3 (1997), 553-556. MR 1485075 | Zbl 0938.54008
[7] Howard P., Keremedis K., Rubin H., Rubin J.E.: Disjoint unions of topological spaces and choice. Math. Logic Quart. 44 (1998), 493-508. MR 1654348 | Zbl 0922.03069
[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). MR 1755811 | Zbl 0993.03059
[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. MR 1847458
[10] Howard P., Rubin J.E.: Consequences of the Axiom of Choice. Math. Surveys and Monographs 59, Amer. Math. Soc., Providence R.I., 1998. MR 1637107 | Zbl 0947.03001
[11] Jech T.: The Axiom of Choice. North-Holland, Amsterdam, 1973. MR 0396271 | Zbl 0259.02052
[12] Keremedis K.: Disasters in topology without the axiom of choice. Arch. Math. Logic, to appear. MR 1867681 | Zbl 1027.03040
[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. MR 1808950 | Zbl 0968.03057
[15] Keremedis K., Tachtsis E.: On Lindelöf metric spaces and weak forms of the axiom of choice. Math. Logic Quart. 46 (2000), 35-44. MR 1736648 | Zbl 0952.03060
[16] Kunen K.: Set Theory, An Introduction to Independence Proofs. North-Holland, Amsterdam, 1983. MR 0756630 | Zbl 0534.03026
[17] Willard S.: General Topology. Addison-Wesley Publ. Co., Reading, MA, 1968. MR 2048350 | Zbl 1052.54001
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