A METRIZATION THEOREM FOR-GENERALIZED ORDERED SPACES

Bennett; details will appear in [BnL]. Most metrization theory for GO spaces ( = gen~ralized ordered spaces; cf. [L] for definitions and termi~ology) is based on Bing's theorem [Bi] that a space is metrizable iff it is collectionwise normal and developable. The theorem which will appear in [BnL] rests on a different base, namely a theorem of Bennett [Bn] which asserts that a space is metrizable iff it is a quasi-developable paracompact p-space. Here we use "p-space " in the sense of Arhangelskii [A] andl we say that a space X is quasi-developable if X has a base B = U{B(n): n ~ l} with the property that whenever p is a point of an open set U, there is an n ~ 1 such that P E St(p,B(n)) c U. (It is important to note that the collec±ions B(n) are not required to be coverings of X.) Other definitions used in this report appear in [BkL]. Th