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Keywords:
weak axioms of choice; pseudometric spaces; metric reflections; complete metric and pseudometric spaces; limit point compact; Alexandroff-Urysohn compact; ultrafilter compact; sequentially compact
weak axioms of choice; pseudometric spaces; metric reflections; complete metric and pseudometric spaces; limit point compact; Alexandroff-Urysohn compact; ultrafilter compact; sequentially compact
Summary:
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 $\mathbf{ZFA}$.
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