| Title:
|
On preimages of ultrafilters in ZF (English) |
| Author:
|
Herrlich, Horst |
| Author:
|
Howard, Paul |
| Author:
|
Keremedis, Kyriakos |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
241-252 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Boolean Prime Ideal Theorem |
| Keyword:
|
weak forms of the axiom of choice |
| Keyword:
|
ultrafilters |
| MSC:
|
06E15 |
| MSC:
|
54D30 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 06604504 |
| idMR:
|
MR3513447 |
| DOI:
|
10.14712/1213-7243.2015.159 |
| . |
| Date available:
|
2016-07-05T15:11:16Z |
| Last updated:
|
2018-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145755 |
| . |
| Reference:
|
[1] Herrlich H., Howard P., Keremedis K.: On extensions of countable filterbases to ultrafilters and ultrafilter compactness.submitted manuscript. |
| Reference:
|
[2] Herrlich H., Keremedis K.: On the metric reflection of a pseudometric space in $\mathbf{ZF}$.Comment. Math. Univ. Carolin. 56 (2015), 77–88. MR 3311579 |
| Reference:
|
[3] Hall E., Keremedis K., Tachtsis E.: The existence of free ultrafilters on $\omega $ does not imply the extension of filters on $\omega $ to ultrafilters.Math. Logic Quart. 59 (2013), 158–267. MR 3100753, 10.1002/malq.201100092 |
| Reference:
|
[4] Howard P., Rubin J. E.: Consequences of the Axiom of Choice.Math. Surveys and Monographs, 59, American Mathematical Society, Providence, R.I., 1998. Zbl 0947.03001, MR 1637107, 10.1090/surv/059 |
| Reference:
|
[5] Jech T.: The Axiom of Choice.North-Holland Publishing Co., Amsterdam-London, 1973. Zbl 0259.02052, MR 0396271 |
| Reference:
|
[6] Keremedis K.: Tychonoff products of two-element sets and some weakenings of the Boolean Prime Ideal Theorem.Bull. Pol. Acad. Sci. Math. 53 (2005), no. 4, 349–359. Zbl 1112.03044, MR 2214925, 10.4064/ba53-4-1 |
| . |
