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: \newline ``{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 ``{for every set $A\not =\emptyset $, every filter $\Cal {F} \subseteq \Cal {P}(A)$ extends to an ultrafilter with a well orderable filter base}'' and \newline ``{every open filter $\Cal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base}''.
Keywords: closed filters, bases for filters, characters of filters, ultrafilters
AMS Subject Classification: Primary 03E25