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In [[mathematics]], the '''compact-open topology''' is a [[topological space|topology]] defined on the set of [[continuous function|continuous maps]] between two [[topological space]]s. The compact-open topology is one of the commonly used topologies on [[function space]]s, and is applied in [[homotopy theory]] and [[functional analysis]].  It was introduced by [[Ralph Fox]] in 1945 [http://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/].
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== Definition ==
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Let ''X'' and ''Y'' be two [[topological space]]s, and let ''C''(''X'',''Y'') denote the set of all [[continuous map]]s between ''X'' and ''Y''. Given a [[compact set|compact subset]] ''K'' of ''X'' and an [[open set|open subset]] ''U'' of ''Y'', let ''V''(''K'',''U'') denote the set of all functions {{nowrap|1=&fnof; &isin; ''C''(''X'',''Y'')}} such that {{nowrap|1=&fnof;(''K'') &sub; ''U''.}} Then the collection of all such ''V''(''K'',''U'') is a [[subbase]] for the compact-open topology on ''C''(''X'',''Y''). (This collection does not always form a [[base (topology)|base]] for a topology on ''C''(''X'',''Y'').)
 
When working in the category of [[compactly generated space]]s, it is common to modify this definition by restricting to the subbase formed from those ''K'' which are the image of a [[compact set|compact]] [[Hausdorff space|Hausdorff]] space. Of course, if ''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, among other useful properties.<ref>{{cite journal |jstor=1995173 |title=Classifying Spaces and Infinite Symmetric Products}}</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}}</ref> The confusion between this definition and the one above is caused by differing usage of the word [[compact set|compact]].
 
== Properties ==
 
* If ''*'' is a one-point space then one can identify ''C''(''*'',''X'') with ''X'', and under this identification the compact-open topology agrees with the topology on ''X''
 
* If ''Y'' is [[T0 space|''T''<sub>0</sub>]], [[T1 space|''T''<sub>1</sub>]], [[Hausdorff space|Hausdorff]], [[regular space|regular]], or [[tychonoff space|Tychonoff]], then the compact-open topology has the corresponding [[separation axiom]].
 
* If ''X'' is Hausdorff and ''S'' is a [[subbase]] for ''Y'', then the collection {''V''(''K'',''U'') : ''U'' in ''S''} is a [[subbase]] for the compact-open topology on ''C''(''X'',''Y'').
 
* If ''Y'' is a [[metric space]] (or more generally, an [[uniform space]]), then the compact-open topology is equal to the [[topology of compact convergence]]. In other words, if ''Y'' is a metric space, then a [[sequence]] {ƒ<sub>''n''</sub>} [[limit (mathematics)|converge]]s to ƒ in the compact-open topology if and only if for every compact subset ''K'' of ''X'', {ƒ<sub>''n''</sub>} converges uniformly to ƒ on ''K''. In particular, if ''X'' is compact and ''Y'' is a uniform space, then the compact-open topology is equal to the topology of [[uniform convergence]].
 
* If ''X'', ''Y'' and ''Z'' are topological spaces, with ''Y'' [[locally compact Hausdorff]] (or even just locally compact [[preregular space|preregular]]), then the [[function composition|composition map]] {{nowrap|1=''C''(''Y'',''Z'') &times; ''C''(''X'',''Y'') &rarr; ''C''(''X'',''Z''),}} given by {{nowrap|1=(&fnof;,''g'') ↦ &fnof; ∘ ''g'',}} is continuous (here all the function spaces are given the compact-open topology and ''C''(''Y'',''Z'') &times; ''C''(''X'',''Y'') is given the [[product topology]]).
 
*If ''Y'' is a locally compact Hausdorff (or preregular) space, then the evaluation map ''e''&nbsp;:&nbsp;''C''(''Y'',''Z'')&nbsp;&times;&nbsp;''Y''&nbsp;→&nbsp;''Z'', defined by ''e''(ƒ,''x'') = ƒ(''x''), is continuous. This can be seen as a special case of the above where ''X'' is a one-point space.
 
* If ''X'' is compact, and if ''Y'' is a metric space with metric ''d'', then the compact-open topology on ''C''(''X'',''Y'') is [[metrisable space|metrisable]], and a metric for it is given by ''e''(ƒ,''g'') = [[supremum|sup]]{''d''(ƒ(''x''), ''g''(''x'')) : ''x'' in ''X''}, for ƒ, ''g'' in ''C''(''X'',''Y'').
 
== Fréchet differentiable functions ==
Let ''X'' and ''Y'' be two [[Banach space]]s defined on the same field, and let <math>\mathcal{C}^m\left(U,Y\right) </math> denote the set of all m-continuously Fréchet-differentiable functions from the open subset <math>U\subseteq X</math> to <math>Y</math>. The compact-open topology is the [[initial topology]] induced by the [[seminorm]]s
:<math>p_{K}\left( f \right) = \sup \{ \| D^{j}f\left( x \right)\|, x\in K, 0\leq j \leq m \}</math>
where <math>D^{0}f\left( x \right) = f\left( x \right)</math>, for each compact subset <math>K\subseteq U</math>.
 
==See also==
* [[Bounded-open topology]]
 
==References==
{{Reflist}}
*{{Cite book|first=J.|last=Dugundji|authorlink=James Dugundji|title=Topology|publisher=Allyn and Becon|year=1966|asin=B000KWE22K}}
* O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) [http://www.math.uu.se/~oleg/topoman.html Textbook in Problems on Elementary Topology].
*{{planetmath reference|id=3976|title=Compact-open topology}}
 
[[Category:General topology]]
[[Category:Topology of function spaces]]

Latest revision as of 00:09, 31 December 2014

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