Sharp bases and weakly uniform bases versus point-countable bases

A base B for a topological space X is said to be sharp if for every x∈X and every sequence (Un)n∈ω of pairwise distinct elements of B with x∈Un for all n the set {⋂i<nUi: n∈ω} forms a base at x. Sharp bases of T0-spaces are weakly uniform. We investigate which spaces with sharp bases or weakly uniform bases have point-countable bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform base, but without a point-countable base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-countable space with a weakly uniform base and at most ℵω isolated points has a point-countable base.