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closed filters; bases for filters; characters of filters; ultrafilters
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\/}''.
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