On the metrizability of spaces with a sharp base

A base B for a space X is said to be sharp if, whenever x∈X and (Bn)n∈ω is a sequence of pairwise distinct element of B each containing x, the collection {⋂j⩽nBj: n∈ω} is a base at the point x. We answer questions raised by Alleche et al. and Arhangelʹskiı̆ et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and [0,1] need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarevʹs for point countable bases.