Weak Hausdorff Gaps and the 𝔭 ≤ t Problem

Abstract

We find topological characterizations of the pseudointersection number 𝔭 and the tower number t of the real line and we show that 𝔭 < t iff there exists a compact separable T2 space X of π-weight < 𝔭 that can be covered by < t nowhere dense sets iff there exists a weak Hausdorff gap of size K < t, i. e., a pair ({A : i ≠ k}, {BJ : j ε K}) C [W]W X [U]W such that A = {Ai : i ε K} is a decreasing tower, B = {Bj : j ε K) is a family of pseudointersections of A, and there is no pseudointersection S of A meeting each member of B in an infinite set.