Sharp bases and metrizability
by
Chris Good
University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Coauthors: Abdul Mohamad, Robin Knight

This is joint work with Abdul Mohamad and Robin Knight.

Alleche et al. [1] say that a base B for space X is sharp if, whenever (B_n) is an injective sequence from B and x in \cap B_n, ( \cap _{j <= n}B_j) is a local base at x. Like developability, having a sharp base is more general than having a uniform base and implies weak developability. Arhangel'skii et al [2] show that a space with a sharp base has a point countable base. Since a pseudocompact space with a uniform base is metrizable but a pseudocompact space with a point countable base need not be, both papers ask whether a pseudocompact space with a sharp base need be metrizable. We show that the answer to this question is no. [1] also asks whether X×[0, 1] has a sharp base whenever X does. Our example shows that the answer to this question is also no since, if X×[0, 1] has a sharp base, then X has a \sigma-point finite base.

We also prove some (simple) metrization theorems for spaces with sharp bases.

[1]

Alleche, Arhangel'skii and Calbrix, Weak developments and sharp bases, Topology Appl. 100, 2000.

[2]

Arhangel'skii, Just, Rezniczenko and Szeptycki, Sharp bases and weakly uniform bases versus point countable bases, ibid.

Date received: July 21, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-64.