Editing Path (topology) (section)

== Homotopy of paths ==
{{Main|Homotopy}}
[[Image:Homotopy between two paths.svg|thumb|right|A homotopy between two paths.]]

Paths and loops are central subjects of study in the branch of [[algebraic topology]] called [[homotopy theory]]. A [[homotopy]] of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.

Specifically, a homotopy of paths, or '''path-homotopy''', in <math>X</math> is a family of paths <math>f_t : [0, 1] \to X</math> indexed by <math>I = [0, 1]</math> such that
* <math>f_t(0) = x_0</math> and <math>f_t(1) = x_1</math> are fixed. 
* the map <math>F : [0, 1] \times [0, 1] \to X</math> given by <math>F(s, t) = f_t(s)</math> is continuous. 
The paths <math>f_0</math> and <math>f_1</math> connected by a homotopy are said to be '''homotopic''' (or more precisely '''path-homotopic''', to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.

The relation of being homotopic is an [[equivalence relation]] on paths in a topological space. The [[equivalence class]] of a path <math>f</math> under this relation is called the '''homotopy class''' of <math>f,</math> often denoted <math>[f].</math>