Certain Special Bases in Ordered Spaces
by
Harold R Bennett
Texas Tech University, Lubbock, TX 79409-1042
Coauthors: David J. Lutzer

In this paper we investigate the role of certain special types of bases for linearly ordered and generalized ordered spaces. We show: a) the generalized ordered space X has a weakly uniform base if and only if X is quasi-developable and has a G_\delta-diagonal; b) the linearly ordered space X has a point-countable base if and only if X is first-countable and has an \omega-in-\omega base, i.e., a base B with the property that for every infinite subset A subset X, the collection { B in B :A subset B } is at most countable; c) the generalized ordered space X is metrizable if and only if it has an open-in-finite base, i.e., a base C such that if U is a nonempty open subset of X, then the collection { C in C :U subset C } is finite. We also give examples to show that these are the sharpest possible results. For example, related to (b) we describe a first-countable generalized ordered space that has no point-countable base and yet has a base B with the property that if a, b are distinct points of X, then the collection { B in B:a, b in B } is countable.

Date received: February 9, 1998