Topology, domain theory and theoretical computer science

Michael W. Mislove
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hp7o5f6pmrg57k2m46s4syavf4" style="color: black;">Topology and its Applications</a> </i> &nbsp;
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 applications of domain-theoretic methods in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. 0 1998 Elsevier Science B.V. All rights reserved. Then it is clear that {fi 1 f E (N -I+?)} is precisely the set of selfmaps of Nl that are strict-i.e.,
more &raquo; ... se that take I to itself. If we endow NI with the discrete topology, then all the functions in (PI, 4 Wl) are continuous. Proposition 2.2. In the compact-open topoZogyon (Nl -+ Wl), the sequence {fnl}nE~ converges to FACl. Proof. Since NI is discrete, the compact-open topology on (NL + N1) is the same as the topology of pointwise convergence, so the result is clear. 0 We also note that by endowing NJ_ with the discrete metric and giving (Wl -+ Nl) the Frechet metric, the proposition remains true for the metric topology on (Nl + Nl). M. W Mislove / Topology and its Applications 89 (1998) 3-59 5 But even though we have convergence of { fn}nE~ to FAC (after suitable identification with {fnl}nEw>, something is lost in this assertion. Namely, the functions {fn}nE~ represent increasing approximations to FAC; indeed, as n increases, so does the amount of information we have about the limit function FAC. In fact, there is a natural order on (N -N) that makes this idea precise. Definition 2.3. Define the extensional order on the space (W -N) by f C g iff dom(f) C dam(g) and gidom(f) = f. where dam(f) = f-'(N). Clearly, in this order any increasing family of partial functions has a supremum-the union. Moreover, the function FAC is the supremum of the family {fn}nE~. Our next goal is to capture this as a "convergence in order". This leads us to the Scott topology. Definition 2.4. Let P be a partially ordered set. ?? P is a complete partial order (cpo for short) if P has a least element-usually denoted I-and if every directed subset of P has a least upper bound in P. Note that a directed set must be nonempty, since the empty set is a finite subset of every set. For example, the family (N -NJ) is a cpo: the nowhere-defined function is the least element, and the supremum of a directed family of functions is just their union. Similarly, we can give RI1 the jut order, whereby z C y if and only if z = I or z = y for all z,y E Nl. This corresponds to the pointwise order on the space (Nl + R?l) of monotone selfmaps of iY1, and in this order the supremum of a directed family of functions is the pointwise supremum, and the constant function with value _L is the least element. Definition 2.5. Let P be a partially ordered set. A subset U C P is Scott open if ?? U=rU={yEP/(%EU)uCy}isanupperset,and ?? (Vi0 C: P directed) u D E U +F D n U # 0. Proposition 2.6. Let P be a partially ordered set. (i) The family of Scott open sets is a To topology on P. (ii) If x, y E P and there is some open set containing x but not containing y, then 11: !z Y. (iii) The following are equivalent: (a) The Scott topology is Tl. (b) P has the discrete order: (c) The Scott topology is T2. (d) The Scott topology is discrete.
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