In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay,[1] is a statement that is independent of the usual axioms of ZFC set theory.
For a cardinal number κ, define the following statement: In this context, a set D is called dense if every element of P has a lower bound in D. For application of ccc, an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order).
MA(2ℵ0) is false: [0, 1] is a separable compact Hausdorff space, and so (P, the poset of open subsets under inclusion, is) ccc.
The two sets combined are also of size 𝔠, and a filter meeting both must simultaneously avoid all points of [0, 1] while containing sets of arbitrarily small diameter.
But a filter F containing sets of arbitrarily small diameter must contain a point in ⋂F by compactness.