On the metric reflection of a pseudometric space in ZF

摘要

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 $mathbf{ZFA}$.