Geodesic
Disasters in metric topology without choice
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 165-174.

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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 {\it the disjoint union of metrizable spaces is normal\/}.
Classification : 03E25, 54A35, 54D20, 54E35, 54E45, 54F05
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. 
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Tachtsis, Eleftherios. Disasters in metric topology without choice. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 165-174. http://geodesic.mathdoc.fr/item/CMUC_2002__43_1_a14/