Four Degrees of Completeness
by
John Mack
University of Kentucky

When C(X, C) is assigned the compact-open topology, it is desirable from the point of view of functional analysis for X to be realcompact and that C(X, C) be complete as a uniform space. This requires that X be both realcompact and compactly generated; conditions which are rarely simultaneously achievable outside of separable metric spaces X. For this reason it is desirable to study uniform space properties that are slightly more general than completeness.

In this note, we consider three uniformity properties that are more general than completeness and then identify the topological properties of the underlying space X which insure that C(X, C) has the corresponding completeness property.

Definition. A uniform space (A, \mu) is complete if each Cauchy net converges. It is countably complete if each countably valued Cauchy net converges and sequentially complete if each Cauchy sequence converges. A topological algebra (A, T) is advertibly complete if each advertibly Cauchy net converges. A Cauchy net {y_\alpha} is advertibly Cauchy if there exists x in A so that x+y_\alpha - x y_\alpha and x+y_\alpha - y_\alphax each converge to 0.

Theorem. For any completely regular space X, C(X, C) is countably complete if and only if X is a k_r^\omega-space.

Theorem. (a) Each open subspace of a k_r^\omega-space is k_r^\omega.

(b) If X is a k_r^\omega-space, then the Hewitt realcompactification \upsilonX is k_r^\omega.

Date received: June 19, 2000

Copyright © 2000 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 # caeu-28.