This article is an extended survey of the ways in which general topology, and more specifically the interconnections between topology and ordered sets, have played a fundamental role in the development of theoretical computer science. The author makes it clear that his attention in writing it was to `attract the attention of topologists to this area'; although the reviewer is not (by this definition) a topologist -- being already well aware of the developments in domain theory which form the central theme of this article -- he believes that the author has done an excellent job of expounding the material to those who have not met it before. In particular, the article provides a useful counterbalance to a recent one in the same journal by \textit{J. W. de Bakker} and \textit{E. P. de Vink} [Denotational models for programming languages: Application of Banach's fixed point theorem, ibid. 85, No. 1-3, 35-52 (1998)], which described the applications of complete metric spaces in denotational semantics, but underplayed the order-theoretical aspects of the subject.