The Smyth completion ( 15] , 16], 18] and 19]) provides a topological foundation for Denotational Semantics. We s h o w that this theory simultaneously provides a topological foundation for the complexity analysis of programs via the new theory of \complexity (distance) spaces". The complexity spaces are shown to be weightable ( 13], 8], 10]) and thus belong to the class of S-completable quasi-uniform spaces ( 19]). We show that the S-completable spaces possess a sequential Smyth completion.

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... applicability of the theory to \Divide & Conquer" algorithms is illustrated by a new proof (based on the Banach theorem) of the fact that mergesort has optimal asymptotic average running time.