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Boolean Prime Ideal Theorem; weak forms of the axiom of choice; ultrafilters
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.
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