  A combinatorial principle CECA is formulated and its equivalence with GCH+
certain weakenings of Box_lambda for singular lambda is proved. CECA is 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 family. This
theorem in turn is used to show the consistency of ``every first countable
T_1-space with a weakly uniform base has a point-countable base.''
