Editing Compact-open topology (section)

=== 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.