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 elements of B each containing x, the collection { ⋂ j≤n Bj : n ∈ ω} is a local base at x. We answer questions raised by Alleche et al. and Arhangel’skĭı 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. The notion of a uniform base was introduced by Alexandroff who proved that a space (by which we mean T1 topological space) is metrizable if and only if it has a uniform base and is collectionwise normal [1]. This result follows from Bing’s metrization theorem since a space has a uniform base if and only if it is metacompact and developable. Recently Alleche, Arhangel’skĭı and Calbrix [2] introduced the notions of sharp base and weak development, which fit very naturally into the hierarchy of such strong base conditions including weakly uniform bases (introduced by Heath and Lindgren [11]) and point countable bases (see Figure 1 below). In this paper we look at the question of when a space, with a sharp base is metrizable. In particular, we show that a pseudocompact space with a sharp base need not be metrizable, but generalize various situations where a space with a sharp base is seen to be metrizable. Definition 1. Let B be a base for a space X. 2000 Mathematics Subject Classification. 54E20, 54E30.