Strongly almost disjoint sets and weakly uniform bases

Abstract

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of ◻ λ \Box _\lambda for singular λ \lambda is proved. CECA is used to show that certain “almost point- > τ >\tau ” families can be refined to point- > τ > \tau families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of “every first countable T 1 T_1 -space with a weakly uniform base has a point-countable base.”