4 - When m > ω₁

Summary

I come at last to results which need the full strength of the cardinal m; that is to say, which involve partially ordered sets which are ccc but may not satisfy Knaster's condition. I divide these into two sections on combinatorics (§§41–42) and two on general topology (§§43–44).

Combinatorics I

I base this section on the result that if m > ω₁ then every ccc partially ordered set satisfies Knaster's condition [41Ab]. This is already enough to prove that the product of ccc spaces is ccc [41E], so that Souslin's hypothesis is true [41D, 41F–G], and enables us to apply the results of §31 to ccc sets [41B–C]. I give a result in the partition calculus [41H] with some important corollaries. I conclude with a version of ‘Devlin's axiom’ [41K] and a description of some principles apparently weaker than m > ω₁ [41L].

Theorem

[m > ω₁] Let P be an upwards-ccc partially ordered set.

1. (a) If is a family in P, there is an uncountable A ξ ω₁ such that {p_ξ:ξ∈A} is upwards-centered in P.

2. (b) P satisfies Knaster's condition upwards.

Proof (a) For ξ, < ω₁ set

Q_ξ = {p:p∈P, ∃η≥ξ such that p ≥ p_η}.

Then each Q_ξ is up-open in P, and Q_ξ ξ Q_η whenever η ≤ ξ. Now there is a ζ < ω₁ such that Q_ξ is cofinal with Q_ξ for every ξ ≥ ζ. P?