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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.
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.
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 {\it the disjoint union of metrizable spaces is normal\/}.
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