Between property (K) and the countable chain condition
William Wistar Comfort, S. Negrepontis · Cambridge University Press eBooks · 1982
We have seen in Chapter 6 that the c.c.c. property (for compact spaces) branches into two stronger and logically independent properties: the property of calibre ω + and the existence of a strictly positive measure. These two properties have in fact a common denominator, stronger than c.c.c: Knaster's property (K) (indeed, property K n for 2 ≤ n < ω). Between the c.c.c. and property (K) there is a significant qualitative difference: the c.c.c. cannot be proved in ZFC to be productive, but property (K) is productive (Theorem 2.2(a)). In this chapter we are concerned with differentiating property (K) from c.c.c, and with properties that lie between the two. The principal results, both assuming the continuum hypothesis and using combinatorial methods, are these: (a) There is a c.c.c space X such that X × X is not a c.c.c space (Theorem 7.13, due to R. Laver and F. Galvin); and (b) there is a productively c.c.c. space that does not have property (K) (Theorem 7.9, due to K. Kunen). In the Notes to this chapter we describe the effect of Martin's axiom on the countable chain properties. Kunen's example 7.1 Lemma . Let α be an infinite cardinal and X a space such that S(X) ≤ cf(α). If { V ξ : ξ, < α} is a set of non-empty open subsets of X such that V ξ′ ⊂ V ξ for ξ < ξ′ < α, then {cl V ξ : ξ < α} stabilizes.