Geodesic
On the metric reflection of a pseudometric space in ZF
Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 1, pp. 77-88.

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We show: (i) The countable axiom of choice $\mathbf{CAC}$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $\mathbf{CMC}$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $\mathbf{AC}$ is equivalent to each one of the statements: (a) a pseudometric space is Alexandroff-Urysohn compact iff its metric reflection is Alexandroff-Urysohn compact, (b) a pseudometric space $\mathbf{X}$ is Alexandroff-Urysohn compact iff its metric reflection is ultrafilter compact. (iv) We show that the statement “The preimage of an ultrafilter extends to an ultrafilter” is not a theorem of $\mathbf{ZFA}$.
DOI : 10.14712/1213-7243.015.107
Classification : 54E35, 54E45
Keywords: weak axioms of choice; pseudometric spaces; metric reflections; complete metric and pseudometric spaces; limit point compact; Alexandroff-Urysohn compact; ultrafilter compact; sequentially compact 
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Herrlich, Horst; Keremedis, Kyriakos. On the metric reflection of a pseudometric space in ZF. Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 1, pp. 77-88. doi : 10.14712/1213-7243.015.107. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.015.107/

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