On the Metrizability of Spaces with a Sharp Base

A base for a space is said to be if, whenever ∈ and () is a sequence of pairwise distinct elements of each containing , the collection {∩ : ∈ } is a local base at . We answer questions raised by Alleche and Arhangel'skii 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 Ponomarev's for point countable bases