Cauchy filters and strong completeness of quasi-uniform spaces

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,U ∨ 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. (Let us recall that the (quasi-metric) complexity space was introduced by Schellekens to study complexity analysis of programs.) We characterize those T0 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 T0 strongly complete quasi-uniform space is Smyth complete. We also show that every T1 strongly complete quasi-uniform space is smallset symmetric, so every T1 strongly complete quasi-metric space is (completely) metrizable. AMS(1991) Subject Classification: 54E15, 54D30, 54E35.