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:
“But then people don’t read literature in order to understand; they read it because they want to re-live the feelings and sensations which they found exciting in the past. Art can be a lot of things; but in actual practice, most of it is merely the mental equivalent of alcohol and cantharides.”
—Aldous Huxley (1894–1963)
“The necessary has never been man’s top priority. The passionate pursuit of the nonessential and the extravagant is one of the chief traits of human uniqueness. Unlike other forms of life, man’s greatest exertions are made in the pursuit not of necessities but of superfluities.”
—Eric Hoffer (1902–1983)