The cohesive property

Abstract

We introduce the concept of cohesive families of neighborhood bases. We thereby obtain conditions necessary and sufficient to ensure that a separable space be second countable, and sufficiency conditions for complete collectionwise normality. As by-products we obtain metrizability criteria. We prove, e.g., that a T 1 {T_1} space is metrizable iff it has a refined development { G n : n ∈ N } \{ {G_n}:n \in N\} such that { B p : p ∈ X } \{ {B_p}:p \in X\} with B p = { St ( p , G n ) : n ∈ N } {B_p} = \{ {\text {St}}(p,{G_n}):n \in N\} is cohesive.