Editing Homotopy (section)

==Formal definition==
[[File:Mug and Torus morph.gif|thumb|right|250px|A homotopy and its inverse, between two [[embedding]]s of the [[torus]] into <math>\mathbb{R}^3</math>: as "the surface of a doughnut" and as "the surface of a coffee mug".  This is also an example of an [[#Isotopy|isotopy]].]]
Formally, a homotopy between two [[continuous function (topology)|continuous function]]s ''f'' and ''g'' from a 
topological space ''X'' to a topological space ''Y'' is defined to be a [[Function of several real variables#Continuity and limit|continuous function]] <math>H: X \times  [0,1] \to Y</math> from the [[product topology|product]] of the space ''X'' with the [[unit interval]] [0,&thinsp;1] to ''Y'' such that <math>H(x,0) = f(x)</math> and <math>H(x,1) = g(x)</math> for all <math>x \in X</math>.

If we think of the second [[parameter]] of ''H'' as time then ''H'' describes a ''continuous deformation'' of ''f'' into ''g'': at time 0 we have the function ''f'' and at time 1 we have the function ''g''. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from ''f'' to ''g'' as the slider moves from 0 to 1, and vice versa.

An alternative notation is to say that a homotopy between two continuous functions <math>f, g: X \to Y</math> is a family of continuous functions <math>h_t: X \to Y</math> for <math>t \in [0,1]</math> such that <math>h_0 = f</math> and <math>h_1 = g</math>, and the [[Map (mathematics)|map]] <math>(x, t) \mapsto h_t(x)</math> is continuous from <math>X \times [0,1]</math> to <math>Y</math>. The two versions coincide by setting <math>h_t(x) = H(x,t)</math>. It is not sufficient to require each map <math>h_t(x)</math> to be continuous.<ref>{{Cite web|url=https://math.stackexchange.com/q/104515 |title=algebraic topology - Path homotopy and separately continuous functions|website=Mathematics Stack Exchange}}</ref>

The animation that is looped above right provides an example of a homotopy between two [[embedding]]s, ''f'' and ''g'', of the torus into {{nowrap|1=''R''<sup>3</sup>}}.  ''X'' is the torus, ''Y'' is {{nowrap|1=''R''<sup>3</sup>}}, ''f'' is some continuous function from the torus to ''R''<sup>3</sup> that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; ''g'' is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape.  The animation shows the image of ''h''<sub>''t''</sub>(X) as a function of the parameter ''t'', where ''t'' varies with time from 0 to 1 over each cycle of the animation loop.  It pauses, then shows the image as ''t'' varies back from 1 to 0, pauses, and repeats this cycle.

===Properties===

Continuous functions ''f'' and ''g'' are said to be homotopic if and only if there is a homotopy ''H'' taking ''f'' to ''g'' as described above. Being homotopic is an [[equivalence relation]] on the set of all continuous functions from ''X'' to ''Y''. 
This homotopy relation is compatible with [[function composition]] in the following sense: if {{nowrap|1=''f''<sub>1</sub>, ''g''<sub>1</sub> : ''X'' → ''Y''}} are homotopic, and {{nowrap|1=''f''<sub>2</sub>, ''g''<sub>2</sub> : ''Y'' → ''Z''}} are homotopic, then their compositions {{nowrap|1=''f''<sub>2</sub>&thinsp;∘&thinsp;''f''<sub>1</sub>}} and {{nowrap|1=''g''<sub>2</sub>&thinsp;∘&thinsp;''g''<sub>1</sub> : ''X'' → ''Z''}} are also homotopic.