On the metrizability of spaces with a sharp base

A base B for a space X is said to be sharp if, whenever x is an element of X and (B-n)(nis an element ofomega) is a sequence of pairwise distinct element of B each containing x, the collection {boolean AND(jless than or equal ton) B-j: n is an element of omega} is a base at the point x. We answer questions raised by Alleche et al. and Arhangel'skii 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 Ponomarev's for point countable bases. (C) 2002 Elsevier Science B.V. All rights reserved.