On the metrizability of spaces with a sharp base

Details On the metrizability of spaces with a sharp base On the metrizability of spaces with a sharp base

2002-04-10

arxiv.org/abs/math/0204127v1

Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 125--134, Topology Atlas, Toronto, 2002

Mathematics

General Topology

10 pages. Reprinted from Topology and its Applications, in press, Chris Good, Robin W. Knight and Abdul M. Mohamad, On the met

Scientific paper

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$'$ski\u{\i} 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.

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