Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH certain weakenings of square(lambda) for singular lambda is proved. CECA i s used to show that certain "almost point-< tau" families can be refined to point-< tau families by removing a small set from each member of the famil y. This theorem in turn is used to show the consistency of "every first cou ntable T-1-space with a weakly uniform base has a point-countable base."