Consequences
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| < 2k 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 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.
Read more about this topic: Martin's Axiom
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