Statement of Martin's Axiom
For any cardinal k we define an statement, denoted by MA(k):
For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P such that |D| ≤ k, there is a filter F on P such that F ∩ d is non-empty for every d in D.
Since it is a theorem of ZFC that MA(c) fails, the Martin's axiom is stated as:
Martin's Axiom (MA): For every k < c, MA(k) holds.
In this case (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). This differs from, for example, the notion of antichain in the context of trees.
MA is simply true. This is known as the Rasiowa–Sikorski lemma.
MA is false: is a compact Hausdorff space, which is separable and so ccc. It has no isolated points, so points in it are nowhere dense, but it is the union of many points.
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