Geodesic
On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 2, pp. 343-353 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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\/}''.
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\/}''.
Classification : 03E25, 54A20, 54A35, 54D10
Keywords: closed filters; bases for filters; characters of filters; ultrafilters 
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     author = {Keremedis, Kyriakos and Tachtsis, Eleftherios},
     title = {On the extensibility of closed filters in {T}$_1$ spaces and the existence of well orderable filter bases},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {343--353},
     year = {1999},
     volume = {40},
     number = {2},
     mrnumber = {1732656},
     zbl = {0977.03025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1999_40_2_a16/}
}
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Keremedis, Kyriakos; Tachtsis, Eleftherios. On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 2, pp. 343-353. http://geodesic.mathdoc.fr/item/CMUC_1999_40_2_a16/