 Scott continuity

In mathematics, given two partially ordered sets P and Q a function between them is Scottcontinuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: sup f(D) = f(sup D).^{[1]}
A subset O of a partially ordered set P is called Scottopen if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have nonempty intersection with O. The Scottopen subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scottcontinuous if and only if it is continuous with respect to the Scott topology.^{[1]}
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.^{[2]}
Scottcontinuous functions show up in the study of the denotational semantics of computer programs.
Contents
Properties
A Scottcontinuous function is always monotonic.
A subset of a partially ordered set is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets.^{[3]}
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T_{0} separation axiom).^{[3]} However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.^{[3]} The Scottopen sets form a complete lattice when ordered by inclusion.^{[4]}
For any topological space satisfying the T_{0} separation axiom, the topology induces an order relation on that space, the specialization order: x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scottopen sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: The specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.^{[3]}
Examples
The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology.^{[4]}
For CPO, the cartesian closed category of complete partial orders, two particularly notable examples of Scottcontinuous functions are curry and apply.^{[5]}
See also
Footnotes
 ^ ^{a} ^{b} Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 0521360625.
 ^ Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill. Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. 274. SpringerVerlag.
 ^ ^{a} ^{b} ^{c} ^{d} Abramsky, S.; Jung, A. (1994). "Domain theory". In Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E.. Handbook of Logic in Computer Science. Vol. III. Oxford University Press. ISBN 019853762X. http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf.
 ^ ^{a} ^{b} Bauer, Andrej and Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science (Cambridge University Press) 19: 757–838. doi:10.1017/S0960129509007695. http://PaulTaylor.EU/ASD/dedras/. Retrieved October 8, 2010.
 ^ Barendregt, H.P. (1984). The Lambda Calculus. NorthHolland. ISBN 0444875085. (See theorems 1.2.13, 1.2.14)
References
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