Editing Currying (section)

=== Function spaces ===
In the theory of [[function space]]s, such as in [[functional analysis]] or [[homotopy theory]], one is commonly interested in [[continuous function]]s between [[topological space]]s. One writes <math>\text{Hom}(X,Y)</math> (the [[Hom functor]]) for the set of ''all'' functions from <math>X</math> to <math>Y</math>, and uses the notation <math>Y^X</math> to denote the subset of continuous functions. Here, <math>\text{curry}</math> is the [[bijection]]

:<math>\text{curry}:\text{Hom}(X\times Y, Z) \to \text{Hom}(X, \text{Hom}(Y,Z)) ,</math>

while uncurrying is the inverse map. If the set <math>Y^X</math> of continuous functions from <math>X</math> to <math>Y</math> is given the [[compact-open topology]], and if the space <math>Y</math> is [[locally compact Hausdorff]], then

:<math>\text{curry} : Z^{X\times Y}\to (Z^Y)^X</math>

is a [[homeomorphism]]. This is also the case when <math>X</math>, <math>Y</math> and <math>Y^X</math> are [[Compactly generated space|compactly generated]],<ref name="may">{{Cite book |last=May |first=Jon Peter |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A concise course in algebraic topology |date=1999 |publisher=University of Chicago Press |isbn=0-226-51183-9 |edition= |series=Chicago lectures in mathematics |location=Chicago, Ill. |pages=39–55 |oclc=41266205}}</ref>{{rp|at=chapter 5}}<ref>{{Cite web |date=28 May 2023 |title=compactly generated topological space |url=https://ncatlab.org/nlab/show/compactly+generated+topological+space |access-date= |website=nLab}}</ref> although there are more cases.<ref>{{Cite journal |last1=Tillotson |first1=J. |last2=Booth |first2=Peter I. |orig-date=Received 2 October 1978, revised 29 June 1979, published 1 May 1980 |title=Monoidal closed, Cartesian closed and convenient categories of topological spaces |url=https://msp.org/pjm/1980/88-1/pjm-v88-n1-p03-s.pdf |journal=Pacific Journal of Mathematics |location=Memorial University of Newfoundland |publisher=Mathematical Sciences Publishers |publication-place=Berkeley, California |publication-date=March 1980 |volume=88 |issue=1 |pages=35–53 |doi=10.2140/pjm.1980.88.35 |issn=0030-8730 |eissn=1945-5844}}</ref><ref>{{Cite web |date=11 August 2023 |title=convenient category of topological spaces |url=https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces |access-date= |website=nLab}}</ref>

One useful corollary is that a function is continuous [[if and only if]] its curried form is continuous. Another important result is that the [[apply|application map]], usually called "evaluation" in this context, is continuous (note that [[eval]] is a strictly different concept in computer science.) That is,

<math>\begin{align} &&\text{eval}:Y^X \times X \to Y \\
                     && (f,x) \mapsto f(x) \end{align}</math>

is continuous when <math>Y^X</math> is compact-open and <math>Y</math> locally compact Hausdorff.<ref name="rotman">{{Cite book |last=Rotman |first=Joseph Jonah |url=https://books.google.com/books?id=waq9mwUmcQgC |title=An introduction to algebraic topology |date=1988 |publisher=Springer-Verlag |isbn=978-0-387-96678-6 |series=Graduate texts in mathematics; 119 |location=New York |chapter=Chapter 11 |oclc=17383909}}</ref> These two results are central for establishing the continuity of [[homotopy]], i.e. when <math>X</math> is the unit interval <math>I</math>, so that <math>Z^{I\times Y} \cong (Z^Y)^I</math> can be thought of as either a homotopy of two functions from <math>Y</math> to <math>Z</math>, or, equivalently, a single (continuous) path in <math>Z^Y</math>.