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:
“Accountability in friendship is the equivalent of love without strategy.”
—Anita Brookner (b. 1938)
“The delicious faces of children, the beauty of school-girls, “the sweet seriousness of sixteen,” the lofty air of well-born, well-bred boys, the passionate histories in the looks and manners of youth and early manhood, and the varied power in all that well-known company that escort us through life,—we know how these forms thrill, paralyze, provoke, inspire, and enlarge us.”
—Ralph Waldo Emerson (1803–1882)