On the extensibility of closed filters in T1 spaces and the existence of well orderable filter bases
نویسندگان
چکیده
We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T1 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 CFE0 = “closed filters in a T0 space (X, T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter F in a T1 space (X, T ) extends to a maximal closed filter with a well orderable filter base”, “for every set A 6= ∅, every filter F ⊆ P(A) extends to an ultrafilter with a well orderable filter base” and “every open filter F in a T1 space (X, T ) extends to a maximal open filter with a well orderable filter base”.
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