Electronic Version 17 (1992), 59-69

P. Fletcher and W. Hunsaker

In 1975, J. R. Isbell [8] defined a uniformity for a locale, and subsequently A. Pultr defined a uniformity for a frame L in terms of covers of L [11,13]. B. Banaschewski introduced the concept of a strong inclusion for a frame [1 and 2] and proved that a frame L admits a strong inclusion if and only if L is compactifiable. In his doctoral thesis [4], J. L. Frith established a one-to-one correspondence between strong inclusions for a frame L and totally bounded covering uniformities for L . In this note we use an alternative characterization of a uniformity for a frame L in terms of order-preserving functions from L to L , which we established in [5], to obtain results analogous to those of Frith [4, p.63-67]. Our methods of proof differ from those of Frith in that his arguments use covering uniformities and for the most part ours do not. One positive aspect of our approach is that it affords an explicit construction of a base for the unique totally bounded frame uniformity associated with a given strong inclusion.

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