On the metric reflection of a pseudometric space in ZF

Herrlich, Horst, and Keremedis, Kyriakos. "On the metric reflection of a pseudometric space in ZF." Commentationes Mathematicae Universitatis Carolinae 56.1 (2015): 77-88. <http://eudml.org/doc/269880>.

@article{Herrlich2015,
abstract = {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 $\mathbf \{ZFA\}$.},
author = {Herrlich, Horst, Keremedis, Kyriakos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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},
language = {eng},
number = {1},
pages = {77-88},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the metric reflection of a pseudometric space in ZF},
url = {http://eudml.org/doc/269880},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Herrlich, Horst
AU - Keremedis, Kyriakos
TI - On the metric reflection of a pseudometric space in ZF
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 1
SP - 77
EP - 88
AB - 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 $\mathbf {ZFA}$.
LA - eng
KW - 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
UR - http://eudml.org/doc/269880
ER -