# On the extensibility of closed filters in T${}_{1}$ spaces and the existence of well orderable filter bases

Kyriakos Keremedis; Eleftherios Tachtsis

Commentationes Mathematicae Universitatis Carolinae (1999)

- Volume: 40, Issue: 2, page 343-353
- ISSN: 0010-2628

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topKeremedis, Kyriakos, and Tachtsis, Eleftherios. "On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 343-353. <http://eudml.org/doc/248403>.

@article{Keremedis1999,

abstract = {We show that the statement CCFC = “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 Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE$_0$ = “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: “every closed filter $\mathcal \{F\}$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \emptyset $, every filter $\mathcal \{F\} \subseteq \mathcal \{P\}(A)$ extends to an ultrafilter with a well orderable filter base” and “every open filter $\mathcal \{F\}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base”.},

author = {Keremedis, Kyriakos, Tachtsis, Eleftherios},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {closed filters; bases for filters; characters of filters; ultrafilters; closed filters; bases for filters; characters of filters; ultrafilters; countable multiple choice; axiom of choice},

language = {eng},

number = {2},

pages = {343-353},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases},

url = {http://eudml.org/doc/248403},

volume = {40},

year = {1999},

}

TY - JOUR

AU - Keremedis, Kyriakos

AU - Tachtsis, Eleftherios

TI - On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1999

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 40

IS - 2

SP - 343

EP - 353

AB - We show that the statement CCFC = “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 Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE$_0$ = “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: “every closed filter $\mathcal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \emptyset $, every filter $\mathcal {F} \subseteq \mathcal {P}(A)$ extends to an ultrafilter with a well orderable filter base” and “every open filter $\mathcal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base”.

LA - eng

KW - closed filters; bases for filters; characters of filters; ultrafilters; closed filters; bases for filters; characters of filters; ultrafilters; countable multiple choice; axiom of choice

UR - http://eudml.org/doc/248403

ER -

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