On preimages of ultrafilters in ZF

Commentationes mathematicae Universitatis Carolinae

Horst Herrlich¹ 

关键词: Boolean Prime Ideal Theorem;    weak forms of the axiom of choice;    ultrafilters;   

DOI  :  10.14712/1213-7243.2015.159

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

We show that given infinite sets $X,Y$ and a function $fX\rightarrow Y$ 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 $fA\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.

【 授权许可】

CC BY