In this paper, we survey the use of order-theoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of order-theoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...

Abstract. For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P) = Id(P) − {∅}. This note obtains various results to the effect that Id(P) is always, and id(P) often, “essentially larger ” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone map) from Id(P) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S). The slightly smaller object id(P) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strengthened to say that the only posets P0 such that for every natural number n there exists a poset Pn with id n (Pn) ∼ = P0 are those having ascending chain condition. On the other hand, a wide class of cases is noted where id(P) is embeddable in P. Counterexamples are given to many variants of the statements proved. 1. Definitions. Recall that a poset P is said to be upward directed if every pair of elements of P is majorized by some common element, and that a downset in P means a subset d such that x ≤ y ∈ d = ⇒ x ∈ d. The downset generated by a subset X ⊆ P will be written P ↓ X = {y ∈ P | ( ∃ x ∈ X) y ≤ x}. A principal downset