Intersection Properties of Open Sets. IIe

Abstract

Abstract. A topological space is called P 2 (P 3 , P <ω ) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection. We show that (i) there are 0-dimensional P <ω spaces of size 2 ω , (ii) there are compact P <ω spaces of size ω 1 , (iii) the existence of a Ψ-like examples for (ii) is independent of ZFC, (iv) it is consistent that 2 ω is as large as you wish but every first countable (and so every compact) P 2 space has cardinality ≤ ω 1 .

Combinatorial Principles from Adding Cohen Reals

“…In fact, for κ = ω 1 , they are all false! To see that D(ω 1 ) (so also C(ω 1 )) is false we may recall that in [7] we have constructed, in ZFC, a separable, first countable P <ω space X of size ω 1 . (A Hausdorff space X is called P <ω if the intersection of finitely many uncountable open subsets of X is always non-empty.)…”

“…there is no partition of ω into finitely many pieces such that each piece is almost disjoint to uncountably many elements of A). If MA ω 1 holds, then there is no strong Luzin gap (see [7,Theorem 3.2]), so in ZFC one can not construct an almost disjoint family A satisfying the requirements of corollary 2.14.…”