| Title:
|
On the metric reflection of a pseudometric space in ZF (English) |
| Author:
|
Herrlich, Horst |
| Author:
|
Keremedis, Kyriakos |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
77-88 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show: (i) The countable axiom of choice $\mathbf{CAC}$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $\mathbf{CMC}$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $\mathbf{AC}$ is equivalent to each one of the statements: (a) a pseudometric space is Alexandroff-Urysohn compact iff its metric reflection is Alexandroff-Urysohn compact, (b) a pseudometric space $\mathbf{X}$ is Alexandroff-Urysohn compact iff its metric reflection is ultrafilter compact. (iv) We show that the statement “The preimage of an ultrafilter extends to an ultrafilter” is not a theorem of $\mathbf{ZFA}$. (English) |
| Keyword:
|
weak axioms of choice |
| Keyword:
|
pseudometric spaces |
| Keyword:
|
metric reflections |
| Keyword:
|
complete metric and pseudometric spaces |
| Keyword:
|
limit point compact |
| Keyword:
|
Alexandroff-Urysohn compact |
| Keyword:
|
ultrafilter compact |
| Keyword:
|
sequentially compact |
| MSC:
|
54E35 |
| MSC:
|
54E45 |
| idZBL:
|
Zbl 06433807 |
| idMR:
|
MR3311579 |
| DOI:
|
10.14712/1213-7243.015.107 |
| . |
| Date available:
|
2015-03-10T17:38:14Z |
| Last updated:
|
2017-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144190 |
| . |
| Reference:
|
[1] Bentley H.L., Herrlich H.: Countable choice and pseudometric spaces.Topology Appl. 85 (1998), 153–164. Zbl 0922.03068, MR 1617460, 10.1016/S0166-8641(97)00138-7 |
| Reference:
|
[2] Blass A.: The model of set theory generated by countably many generic reals.J. Symbolic Logic 46 (1981), 732–752. Zbl 0482.03022, MR 0641487, 10.2307/2273223 |
| 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] Herrlich H.: Axiom of Choice.Lecture Notes in Mathematics, 1876, Springer, New York, 2006. Zbl 1102.03049, MR 2243715 |
| Reference:
|
[5] Howard P., Keremedis K., Rubin H., Stanley A.: Compactness in countable Tychonoff products and choice.Math. Logic Quart. 46 (2000), 3–16. Zbl 0942.54006, MR 1736645, 10.1002/(SICI)1521-3870(200001)46:1<3::AID-MALQ3>3.0.CO;2-E |
| Reference:
|
[6] 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:
|
[7] Keremedis K.: On the relative strength of forms of compactness of metric spaces and their countable productivity in $\mathbf{ZF}$.Topology Appl. 159 (2012), 3396–3403. MR 2964853, 10.1016/j.topol.2012.08.003 |
| Reference:
|
[8] Munkres J.R.: Topology.Prentice-Hall, New Jersey, 1975. Zbl 0951.54001, MR 0464128 |
| . |
