Editing Homotopy (section)

== Examples ==
* If <math>f, g: \R \to \R^2</math> are given by <math>f(x) := \left(x, x^3\right)</math> and <math>g(x) = \left(x, e^x\right)</math>, then the map <math>H: \mathbb{R} \times [0, 1] \to \mathbb{R}^2</math> given by <math>H(x, t) = \left(x, (1 - t)x^3 + te^x\right)</math> is a homotopy between them.
* More generally, if <math>C \subseteq \mathbb{R}^n</math> is a [[convex set|convex]] subset of [[Euclidean space]] and <math>f, g: [0, 1] \to C</math> are [[path (topology)|paths]] with the same endpoints, then there is a '''linear homotopy'''<ref>{{Cite book|title=Algebraic topology|last=Allen.|first=Hatcher|date=2002|publisher=Cambridge University Press|isbn=9780521795401|location=Cambridge|pages=185|oclc=45420394}}</ref> (or '''straight-line homotopy''') given by 
*: <math>\begin{align}
  H: [0, 1] \times [0, 1] &\longrightarrow C \\
                   (s, t) &\longmapsto (1 - t)f(s) + tg(s).
\end{align}</math>
* Let <math>\operatorname{id}_{B^n}:B^n\to B^n</math> be the [[identity function]] on the unit ''n''-[[ball (mathematics)|disk]]; i.e. the set <math>B^n := \left\{x\in\mathbb{R}^n: \|x\| \leq 1\right\}</math>. Let <math>c_{\vec{0}}: B^n \to B^n</math> be the [[constant function]] <math>c_\vec{0}(x) := \vec{0}</math> which sends every point to the [[Origin (mathematics)|origin]]. Then the following is a homotopy between them:
*: <math>\begin{align}
  H: B^n \times [0, 1] &\longrightarrow B^n \\
                (x, t) &\longmapsto (1 - t)x.
\end{align}</math>