Editing Compact-open topology (section)

== Definition ==
Let {{mvar|X}} and {{mvar|Y}} be two [[topological space]]s, and let {{math|''C''(''X'', ''Y'')}} denote the set of all [[continuous map]]s between {{mvar|X}} and {{mvar|Y}}. Given a [[compact set|compact subset]] {{mvar|K}} of {{mvar|X}} and an [[open set|open subset]] {{mvar|U}} of {{mvar|Y}}, let {{math|''V''(''K'', ''U'')}} denote the set of all functions {{math|&thinsp;''f''&thinsp; ∈ ''C''(''X'', ''Y'')}} such that {{math|&thinsp;''f''&thinsp;(''K'') ⊆ ''U''.}} In other words, <math>V(K, U) = C(K, U) \times_{C(K, Y)} C(X, Y)</math>. Then the collection of all such {{math|''V''(''K'', ''U'')}} is a [[subbase]] for the compact-open topology on {{math|''C''(''X'', ''Y'')}}. (This collection does not always form a [[base (topology)|base]] for a topology on {{math|''C''(''X'', ''Y'')}}.)

When working in the [[category (mathematics)|category]] of [[compactly generated space]]s, it is common to modify this definition by restricting to the subbase formed from those {{mvar|K}} that are the image of a [[compact set|compact]] [[Hausdorff space]]. Of course, if {{mvar|X}} is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of [[Weak Hausdorff space|compactly generated weak Hausdorff]] spaces to be [[Cartesian closed category|Cartesian closed]], among other useful properties.<ref>{{cite journal |jstor=1995173 |title=Classifying Spaces and Infinite Symmetric Products | pages=273–298|last1=McCord |first1=M. C. |journal=Transactions of the American Mathematical Society |year=1969 |volume=146 |doi=10.1090/S0002-9947-1969-0251719-4 |doi-access=free }}</ref><ref>{{cite web |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A Concise Course in Algebraic Topology}}</ref><ref>{{cite web |url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |title=Compactly Generated Spaces |access-date=2012-01-14 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |url-status=dead }}</ref> The confusion between this definition and the one above is caused by differing usage of the word [[compact set|compact]].

If {{mvar|X}} is locally compact, then <math> X \times - </math> from the category of topological spaces always has a right adjoint <math> Hom(X, -) </math>. This adjoint coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed as taking the adjoint of the product in the category of compactly generated spaces instead of the category of topological spaces, which ensures that the right adjoint always exists.