Geodesic
On preimages of ultrafilters in ZF
Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 2, pp. 241-252.

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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.
DOI : 10.14712/1213-7243.2015.159
Classification : 06E15, 54D30, 54E35
Keywords: Boolean Prime Ideal Theorem; weak forms of the axiom of choice; ultrafilters 
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Herrlich, Horst; Howard, Paul; Keremedis, Kyriakos. On preimages of ultrafilters in ZF. Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 2, pp. 241-252. doi : 10.14712/1213-7243.2015.159. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.159/

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