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Disasters in metric topology without choice. (English) Zbl 1072.03030
Summary: 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.

##### MSC:
 03E25 Axiom of choice and related propositions 54A35 Consistency and independence results in general topology 54E35 Metric spaces, metrizability 54E45 Compact (locally compact) metric spaces
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