Weak duals and neighbourhood assignments

Abstract

Given a topological property (or a class) P, the class P′ consists of spaces X such that for any neighbourhood assignment ϕ on X, there exists a subspace Y⊂X with property P for which ϕ(Y)=⋃{ϕ(y):y∈Y} is dense in X. The class P′ are called the weak dual of P or weakly dually P (with respect to neighbourhood assignments). We establish that DCCC is weakly self-dual in the class of weakly regular spaces. If P∈{weakly Lindelöf,CCC,separable}, then P is weakly self-dual in the class of Baire developable spaces. By using Erdös–Radó's theorem, we also prove that: (1) If X is a Baire, weakly dually CCC Hausdorff space with a rank 2-diagonal, then X has cardinality at most 2ω; (2) If X is a Baire, weakly dually DCCC Hausdorff space with a rank 3-diagonal, then X has cardinality at most 2ω; (3) If X is a weakly dually DCCC Hausdorff space with a rank 4-diagonal, then X has cardinality at most 2ω.