Martin's axiom

For any cardinal 𝛋, consider the following statement:

MA(𝛋)

For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense subsets of P such that |D| ≤ 𝛋, there is a filter F on P such that F ∩ d is non-empty for every d in D.

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(ℵ₀) is simply true — the Rasiowa–Sikorski lemma. MA(2^ℵ₀) is false: [0, 1] is a separable compact Hausdorff space, and so (P, the poset of open subsets under inclusion, is) ccc. But now consider the following two size-2^ℵ₀=c families of dense sets in P: no x∈[0, 1] is isolated, and so each x defines the dense subset {S : x∉S}. And each r∈(0, 1], defines the dense subset {S : diam(S)<r}. The two families 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. (See also § Equivalent forms of MA(κ).)

Martin's axiom is then that MA(κ) holds "as long as possible":

Martin's axiom (MA)

For every 𝛋 < c, MA(𝛋) holds.

The following statements are equivalent to MA(𝛋):

• If X is a compact Hausdorff topological space that satisfies the ccc then X is not the union of 𝛋 or fewer nowhere dense subsets.
• If P is a non-empty upwards ccc poset and Y is a family of cofinal subsets of P with |Y| ≤ 𝛋 then there is an upwards-directed set A such that A meets every element of Y.
• Let A be a non-zero ccc Boolean algebra and F a family of subsets of A with |F| ≤ 𝛋. Then there is a boolean homomorphism φ: A → Z/2Z such that for every X in F either there is an a in X with φ(a) = 1 or there is an upper bound b for X with φ(b) = 0.

Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences:

• The union of 𝛋 or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null. In particular, the union of 𝛋 or fewer subsets of R of Lebesgue measure 0 also has Lebesgue measure 0.
• A compact Hausdorff space X with |X| < 2^𝛋 is sequentially compact, i.e., every sequence has a convergent subsequence.
• No non-principal ultrafilter on N has a base of cardinality < 𝛋.
• Equivalently for any x in βN\N we have 𝜒(x) ≥ 𝛋, where 𝜒 is the character of x, and so 𝜒(βN) ≥ 𝛋.
• MA(ℵ₁) implies that a product of ccc topological spaces is ccc (this in turn implies there are no Suslin lines).
• MA + ¬CH implies that there exists a Whitehead group that is not free; Shelah used this to show that the Whitehead problem is independent of ZFC.

• Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum.
• Sheldon W. Davis has suggested in his book that Martin's axiom is motivated by the Baire category theorem.[2]

• Fremlin, David H. (1984). Consequences of Martin's axiom. Cambridge tracts in mathematics, no. 84. Cambridge: Cambridge University Press. ISBN 0-521-25091-9.
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.