homotopy

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Given two topological spaces $X\,$ and $Y\,$ , and two continuous maps $\alpha_1\,$ and $\alpha_2\,$ from $X\,$ :

$\alpha_1 : X \to Y\,$ ,

$\alpha_2 : X \to Y\,$ .

The two maps are said to be homotopic if $\alpha_1\,$ can be continuously deformed into $\alpha_2\,$ :

$H : X \times [0,1] \to Y\,$ , $H\,$ is continuous,

where

$H(x,0) = \alpha_1(x)\,$ ,

$H(x,1) = \alpha_2(x)\,$ .

Homotopy is an equivalence relation and divides the space of continuous maps from $X\,$ to $Y\,$ , denoted $C(X,Y)\,$ , into equivalence classes. Due to the continuity of $H\,$ , the homotopy equivalence classes are topological invariants of the pair $(X,Y)\,$ .