On preimages of ultrafilters in ZF

Herrlich, Horst, Howard, Paul, and Keremedis, Kyriakos. "On preimages of ultrafilters in ZF." Commentationes Mathematicae Universitatis Carolinae 57.2 (2016): 241-252. <http://eudml.org/doc/280130>.

@article{Herrlich2016,
abstract = {We show that given infinite sets $X,Y$ and a function $f:X\rightarrow Y$ which is onto and $n$-to-one for some $n\in \mathbb \{N\}$, the preimage of any ultrafilter $\mathcal \{F\}$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $\mathcal \{M\}$ with a set of atoms $A$ and a finite-to-one onto function $f:A\rightarrow \omega $ such that for each free ultrafilter of $\omega $ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $\mathcal \{M\}$ there exists an ultrafilter compact pseudometric space $\mathbf \{X\}$ such that its metric reflection $\mathbf \{X\}^\{\ast \}$ is not ultrafilter compact.},
author = {Herrlich, Horst, Howard, Paul, Keremedis, Kyriakos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Boolean Prime Ideal Theorem; weak forms of the axiom of choice; ultrafilters},
language = {eng},
number = {2},
pages = {241-252},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On preimages of ultrafilters in ZF},
url = {http://eudml.org/doc/280130},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Herrlich, Horst
AU - Howard, Paul
AU - Keremedis, Kyriakos
TI - On preimages of ultrafilters in ZF
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 2
SP - 241
EP - 252
AB - We show that given infinite sets $X,Y$ and a function $f:X\rightarrow Y$ which is onto and $n$-to-one for some $n\in \mathbb {N}$, the preimage of any ultrafilter $\mathcal {F}$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $\mathcal {M}$ with a set of atoms $A$ and a finite-to-one onto function $f:A\rightarrow \omega $ such that for each free ultrafilter of $\omega $ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $\mathcal {M}$ there exists an ultrafilter compact pseudometric space $\mathbf {X}$ such that its metric reflection $\mathbf {X}^{\ast }$ is not ultrafilter compact.
LA - eng
KW - Boolean Prime Ideal Theorem; weak forms of the axiom of choice; ultrafilters
UR - http://eudml.org/doc/280130
ER -