Editing Compact-open topology

{{Short description|Type of topology}}
In [[mathematics]], the '''compact-open topology''' is a [[topological space|topology]] defined on the [[set (mathematics)|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.<ref>{{Cite journal|url=https://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/|doi = 10.1090/S0002-9904-1945-08370-0|title = On topologies for function spaces|year = 1945|last1 = Fox|first1 = Ralph H.|journal = Bulletin of the American Mathematical Society|volume = 51|issue = 6|pages = 429–433|doi-access = free}}</ref>

If the [[codomain]] of the [[function (mathematics)|functions]] under consideration has a [[uniform space|uniform structure]] or a [[metric space|metric structure]] then the compact-open topology is the "topology of [[uniform convergence]] on [[compact set]]s." That is to say, a [[sequence]] of functions [[limit (mathematics)|converges]] in the compact-open topology precisely when it converges uniformly on every compact subset of the [[Domain of a function|domain]].<ref>{{cite book|last1=Kelley|first1=John L.|title=General topology|date=1975|publisher=Springer-Verlag|page=230}}</ref>

== 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.

== Properties ==
* If {{math|*}} is a one-point space then one can identify {{math|''C''(*, ''Y'')}} with {{mvar|Y}}, and under this identification the compact-open topology agrees with the topology on {{mvar|Y}}. More generally, if {{mvar|X}} is a [[discrete space]], then {{math|''C''(''X'', ''Y'')}} can be identified with the [[cartesian product]] of {{math|{{!}}''X''{{!}}}} copies of {{mvar|Y}} and the compact-open topology agrees with the [[product topology]].
* If {{mvar|Y}} is {{math|[[T0 space|''T''<sub>0</sub>]]}}, {{math|[[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 {{mvar|X}} is Hausdorff and {{mvar|S}} is a [[subbase]] for {{mvar|Y}}, then the collection {{math|{''V''(''K'',&nbsp;''U'') : ''U'' ∈ ''S'', ''K'' compact} }}is a [[subbase]] for the compact-open topology on {{math|''C''(''X'', ''Y'')}}.<ref>{{cite journal |jstor=2032279 |title=Spaces of Mappings on Topological Products with Applications to Homotopy Theory |author=Jackson, James R. |journal=Proceedings of the American Mathematical Society |year=1952 |volume=3 |issue=2 |pages=327–333 |doi=10.1090/S0002-9939-1952-0047322-4 | url=https://www.ams.org/journals/proc/1952-003-02/S0002-9939-1952-0047322-4/S0002-9939-1952-0047322-4.pdf|doi-access=free }}</ref>
* If {{mvar|Y}} is a [[metric space]] (or more generally, a [[uniform space]]), then the compact-open topology is equal to the [[topology of compact convergence]]. In other words, if {{mvar|Y}} is a metric space, then a sequence {{math|{&thinsp;''f''<sub>''n''</sub>&thinsp;} }}[[limit (mathematics)|converge]]s to {{math|&thinsp;''f''&thinsp;}} in the compact-open topology if and only if for every compact subset {{mvar|K}} of {{mvar|X}}, {{math|{&thinsp;''f''<sub>''n''</sub>&thinsp;} }}converges uniformly to {{math|&thinsp;''f''&thinsp;}} on {{mvar|K}}. If {{mvar|X}} is compact and {{mvar|Y}} is a uniform space, then the compact-open topology is equal to the topology of [[uniform convergence]].
* If {{math|''X'', ''Y''}} and {{mvar|Z}} are topological spaces, with {{mvar|Y}} [[locally compact Hausdorff]] (or even just locally compact [[preregular space|preregular]]), then the [[function composition|composition map]] {{math|''C''(''Y'', ''Z'')&thinsp;×&thinsp;''C''(''X'', ''Y'') → ''C''(''X'', ''Z''),}} given by {{math|(&thinsp;''f''&thinsp;, ''g'') ↦ &thinsp;''f''&thinsp;∘&thinsp;''g'',}} is continuous (here all the function spaces are given the compact-open topology and {{math|''C''(''Y'', ''Z'')&thinsp;×&thinsp;''C''(''X'', ''Y'')}} is given the [[product topology]]).
*If {{mvar|X}} is a locally compact Hausdorff (or preregular) space, then the evaluation map {{math|''e'' : ''C''(''X'', ''Y'') × ''X'' → ''Y''}}, defined by {{math|''e''(&thinsp;''f''&thinsp;, ''x'') {{=}} &thinsp;''f''&thinsp;(''x'')}}, is continuous. This can be seen as a special case of the above where {{mvar|X}} is a one-point space.
* If {{mvar|X}} is compact, and {{mvar|Y}} is a metric space with [[metric (mathematics)|metric]] {{mvar|d}}, then the compact-open topology on {{math|''C''(''X'', ''Y'')}} is [[metrizable space|metrizable]], and a metric for it is given by {{math|''e''(&thinsp;''f''&thinsp;, ''g'') {{=}} [[supremum|sup]]{''d''(&thinsp;''f''&thinsp;(''x''), ''g''(''x'')) : ''x'' in ''X''},}} for {{math|&thinsp;''f''&thinsp;, ''g''}} in {{math|''C''(''X'', ''Y'')}}. More generally, if {{mvar|X}} is [[Hemicompact_space|hemicompact]], and {{mvar|Y}} metric, the compact-open topology is metrizable by the [[Hemicompact_space#Applications|construction linked here]]. 

=== Applications ===
The compact open topology can be used to topologize the following sets:<ref name=":0">{{Cite book|last1=Fomenko|first1=Anatoly|title=Homotopical Topology|last2=Fuchs|first2=Dmitry|edition=2nd|pages=20–23}}</ref>

* <math>\Omega(X,x_0) = \{ f: I \to X \mid f(0) = f(1) = x_0 \}</math>, the [[loop space]] of <math>X</math> at <math>x_0</math>,
* <math>E(X, x_0, x_1) = \{ f: I \to X \mid f(0) = x_0 \text{ and } f(1) = x_1 \}</math>,
* <math>E(X, x_0) = \{ f: I \to X \mid f(0) = x_0 \}</math>.

In addition, there is a [[Homotopy#Homotopy equivalence|homotopy equivalence]] between the spaces <math>C(\Sigma X, Y) \cong C(X, \Omega Y)</math>.<ref name=":0" /> The topological spaces <math>C(X,Y)</math> are useful in homotopy theory because they can be used to form a topological space and a model for the homotopy type of the ''set'' of homotopy classes of maps{{clarify|reason=This sentence is not very comprehensible. Also, the following math seems notationally confused|date=March 2025}}
<math display=block>\pi(X,Y) = \{[f]: X \to Y \mid f \text{ is a homotopy class}\}.</math>
This is because <math>\pi(X,Y)</math> is the set of path components in <math>C(X,Y)</math>{{endash}}that is, there is an [[isomorphism]] of sets
<math display=block>\pi(X,Y) \to C(I, C(X, Y))/{\sim},</math>
where <math>\sim</math> is the homotopy equivalence.

== Fréchet differentiable functions ==
Let {{mvar|X}} and {{mvar|Y}} be two [[Banach space]]s defined over the same [[field (mathematics)|field]], and let {{math|''C<sup>&thinsp;m</sup>''(''U'', ''Y'')}} denote the set of all {{mvar|m}}-continuously [[Fréchet derivative|Fréchet-differentiable]] functions from the open subset {{math|''U'' ⊆ ''X''}} to {{mvar|Y}}. The compact-open topology is the [[initial topology]] induced by the [[seminorm]]s

:<math>p_{K}(f) = \sup \left\{ \left\| D^j f(x) \right\| \ : \ x \in K, 0 \leq j \leq m \right\}</math>

where {{math|''D''<sup>0</sup>&thinsp;''f''&thinsp;(''x'') {{=}} &thinsp;''f''&thinsp;(''x'')}}, for each compact subset {{math|''K'' ⊆ ''U''}}.{{clarification needed|date=February 2022|reason=Is this original research showing that this definition is equivalent in this special case to the general definition given above? Or is it a definition copied from an external reference, in which case that reference should be cited?}}

== See also ==

* [[Topology of uniform convergence]]
* {{annotated link|Uniform convergence}}

== References ==
{{reflist}}
* {{Cite book|first=J.|last=Dugundji|author-link=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|urlname=CompactOpenTopology|title=Compact-open topology}}
*  [http://groupoids.org.uk/topgpds.html Topology and Groupoids Section 5.9 ] Ronald Brown, 2006

[[Category:General topology]]
[[Category:Topology of function spaces]]