**MSC:**- 54E15 Uniform structures and generalizations
- 54D30 Compactness
- 54E35 Metric spaces, metrizability

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.