Equivalent Forms of MA(k)
The following statements are equivalent to Martin's axiom:
- If X is a compact Hausdorff topological space which 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.
Read more about this topic: Martin's Axiom
Famous quotes containing the words equivalent and/or forms:
“Divorce is the psychological equivalent of a triple coronary bypass.”
—Mary Kay Blakely (20th century)
“One way to think about play, is as the process of finding new combinations for known things—combinations that may yield new forms of expression, new inventions, new discoveries, and new solutions....It’s exactly what children’s play seems to be about and explains why so many people have come to think that children’s play is so important a part of childhood—and beyond.”
—Fred Rogers (20th century)