On the metrizability of spaces with a sharp base

Abstract

A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n{)}_{n\in \omega }$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{{\bigcap }_{j\le n}{B}_j:n\in \omega \}$ is a local base at $x$. We answer questions raised by Alleche et al. and Arhangel${}^{\prime }$ski\uı 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.

10 pages. 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