Editing Currying (section)

=== Domain theory ===
In [[order theory]], that is, the theory of [[lattice (order)|lattices]] of [[partially ordered set]]s, <math>\text{curry}</math> is a [[continuous function]] when the lattice is given the [[Scott topology]].<ref>{{Cite book |last=Barendregt |first=Hendrik Pieter |title=The lambda calculus: its syntax and semantics |date=1984 |publisher=North-Holland, an imprint of Elsevier |isbn=978-0-444-87508-2 |edition=Rev. |series=Studies in logic and the foundations of mathematics |volume=103 |location= |chapter=Theorems 1.2.13, 1.2.14}}</ref> Scott-continuous functions were first investigated in the attempt to provide a semantics for [[lambda calculus]] (as ordinary set theory is inadequate to do this). More generally, Scott-continuous functions are now studied in [[domain theory]], which encompasses the study of [[denotational semantics]] of computer algorithms. Note that the Scott topology is quite different than many common topologies one might encounter in the [[category of topological spaces]]; the Scott topology is typically [[final topology|finer]], and is not [[sober space|sober]].

The notion of continuity makes its appearance in [[homotopy type theory]], where, roughly speaking, two computer programs can be considered to be homotopic, i.e. compute the same results, if they can be "continuously" [[code refactoring|refactored]] from one to the other.