Martin's axiom

For any cardinal k, we define a 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(${\displaystyle {\mathfrak {c}}}$) fails, Martin's axiom is stated as:

Martin's axiom (MA): For every k < ${\displaystyle {\mathfrak {c}}}$, 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(${\displaystyle \aleph _{0}}$) is simply true. This is known as the Rasiowa–Sikorski lemma.

MA(${\displaystyle 2^{\aleph _{0}}}$) is false: [0, 1] 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 ${\displaystyle 2^{\aleph _{0}}={\mathfrak {c}}}$ many points. (See the condition equivalent to ${\displaystyle \operatorname {MA} ({\mathfrak {c}})}$ below.)

The following statements are equivalent to MA(k):

• If X is a compact Hausdorff topological space that satisfies the ccc then X is not the union of k 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| ≤ k 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| ≤ k. 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 k or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null. In particular, the union of k or fewer subsets of R of Lebesgue measure 0 also has Lebesgue measure 0.
• A compact Hausdorff space X with |X| < 2^k is sequentially compact, i.e., every sequence has a convergent subsequence.
• No non-principal ultrafilter on N has a base of cardinality < k.
• Equivalently for any x in βN\N we have χ(x) ≥ k, where χ is the character of x, and so χ(βN) ≥ k.
• MA(${\displaystyle \aleph _{1}}$) 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 (Davis 2005, p. 29).

• Davis, Sheldon W. (2005). Topology. McGraw Hill. ISBN 0-07-291006-2.CS1 maint: ref=harv (link)
• 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.
• Martin, D. A.; Solovay, R. M. (1970), "Internal Cohen extensions.", Ann. Math. Logic, 2 (2): 143–178, doi:10.1016/0003-4843(70)90009-4, MR 0270904