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