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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.

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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