| Title:
|
On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases (English) |
| Author:
|
Keremedis, Kyriakos |
| Author:
|
Tachtsis, Eleftherios |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
343-353 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that the statement CCFC = ``{\it the character of a maximal free filter $F$ of closed sets in a $T_1$ space $(X,T)$ is not countable\/}'' is equivalent to the {\it Countable Multiple Choice Axiom\/} CMC and, the axiom of choice AC is equivalent to the statement CFE$_0$ = ``{\it closed filters in a $T_0$ space $(X,T)$ extend to maximal closed filters\/}''. We also show that AC is equivalent to each of the assertions: \newline ``{\it every closed filter $\Cal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base\/}'', \newline ``{\it for every set $A\neq \emptyset $, every filter $\Cal {F} \subseteq \Cal {P}(A)$ extends to an ultrafilter with a well orderable filter base\/}'' and \newline ``{\it every open filter $\Cal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base\/}''. (English) |
| Keyword:
|
closed filters |
| Keyword:
|
bases for filters |
| Keyword:
|
characters of filters |
| Keyword:
|
ultrafilters |
| MSC:
|
03E25 |
| MSC:
|
54A20 |
| MSC:
|
54A35 |
| MSC:
|
54D10 |
| idZBL:
|
Zbl 0977.03025 |
| idMR:
|
MR1732656 |
| . |
| Date available:
|
2009-01-08T18:52:56Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119091 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| . |
