## Eleftherios Tachtsis

*Disasters in metric topology without choice *

Comment.Math.Univ.Carolinae 43,1 (2002) 165-174. **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}.

**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.

**AMS Subject Classification:** 03E25, 54A35, 54D20, 54E35, 54E45, 54F05

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