Topology Atlas
Document # ppae-12

## On the metrizability of spaces with a sharp base

### Chris Good, Robin W. Knight and Abdul M. Mohamad

Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 125-134
A base *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 *B* each containing x, the collection
{ \cap _{j <= n}B_{j}: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.

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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`

Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.