Salvador Romaguera, Michel Schellekens
Cauchy filters and strong completeness of quasi-uniform spaces
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 54, 69-79 (2000)

MSC:

54E15 Uniform structures and generalizations

54D30 Compactness

54E35 Metric spaces, metrizability

Abstract: We introduce and study the notions of a strongly completable and of a strongly complete
quasi-uniform space. A quasi-uniform space $(X, U)$ is said to be strongly complete if every Cauchy
filter (in the sense of Sieber and Pervin) clusters in the uniform space $(X,{\cal U}\vee {\cal
U}^{-1})$. An interesting motivation for the study of this notion of completeness is the fact,
proved here, that the quasi-uniformity induced by the complexity space is strongly complete but not
Corson complete. We recall that the (quasi-metric) complexity space was introduced by Schellekens
to study complexity analysis of programs. We characterize those quasi-uniform space that are
strongly completable and show that a quasi-uniform space is strongly complete if and only if it is
bicomplete and strongly completable. We observe that every $T_0$ strongly complete quasi-uniform space
is Smyth complete. We also show that every $T_1$ strongly complete quasi-uniform space is small-set
symmetric, so every $T_1$ strongly complete quasi-metric space is (completely) metrizable.\\[2ex]
KEY WORDS. Cauchy filter, strongly complete, Corson complete, Smyth complete, bicomplete, small
set-symmetric, complexity space.
Keywords: Uniform structures and generalizations, Compactness, Metric spaces, metrizability