Editing Compact-open topology (section)

== 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.