A base B for a space X is said to be sharp if, whenever x in X and (Bn)n in \omega is a sequence of pairwise distinct elements of B each containing x, the collection { \cap j <= nBj:n in \omega} is a local base at x. We answer questions raised by Alleche et al. and Arhangel'ski 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.
Mathematics Subject Classification. 54E20 54E30.
Keywords. Tychonoff space, pseudocompact; special bases; sharp base;
metrizability.
Comments. This article is reprinted from Topology and its Applications, in press, Chris Good, Robin W. Knight and Abdul M. Mohamad, On the metrizability of spaces with a sharp base, Copyright (2002), with permission from Elsevier Science. http://www.elsevier.com/locate/topol http://sciencedirect.com