Weight = proximity weight
by
Richard E. Hodel
Duke University

Smirnov showed that there is a one-to-one correspondence between totally bounded uniform spaces and proximity spaces; moreover, Hunsaker and Lindgren have extended this result to quasi-uniform and quasi-proximity spaces. We prove cardinal function versions of these two theorems.

Definition. Let X be a set and let \delta be a quasi-proximity on X (more generally, any binary relation on P(X)). A base for \delta is a collection of ordered pairs

B = {<C, D>: C \not\deltaD }

such that if A \not\deltaB, then there exists <C, D> in B such that A subset or equal C and B subset or equal D. The quasi-proximity weight of a space X, denoted q\delta(X), is the smallest infinite cardinal \kappa such that there is a quasi-proximity on X that induces the toplogy of X and has a base of cardinality at most \kappa. There is a similar definition of \delta(X), the proximity weight of a completely regular space X.

Theorem 1. For any space X, w(X) = q\delta(X).

Theorem 2. For any completely regular space X, w(X) = \delta(X).

Date received: March 10, 1997

Copyright © 1997 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 # caam-51.