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:
“Nobody can deny but religion is a comfort to the distressed, a cordial to the sick, and sometimes a restraint on the wicked; therefore whoever would argue or laugh it out of the world without giving some equivalent for it ought to be treated as a common enemy.”
—Mary Wortley, Lady Montagu (1689–1762)
“Culture’s essential service to a religion is to destroy intellectual idolatry, the recurrent tendency in religion to replace the object of its worship with its present understanding and forms of approach to that object.”
—Northrop Frye (b. 1912)