Homotopy for Maps vs Paths

Homotopy (of maps)

A homotopy is a family of maps f_t: X\to Y, t\in I, such that the associated map F:X\times I\to Y given by F(x,t)=f_t(x) is continuous. Two maps f_0, f_1:X\to Y are called homotopic, denoted f_0\simeq f_1, if there exists a homotopy f_t connecting them.

Homotopy of paths

A homotopy of paths in a space X is a family f_t: I\to X, 0\leq t\leq 1, such that

(i) The endpoints f_t(0)=x_0 and f_t(1)=x_1 are independent of t.
(ii) The associated map F:I\times I\to X defined by F(s,t)=f_t(s) is continuous.

When two paths f_0 and f_1 are connected in this way by a homotopy f_t, they are said to be homotopic. The notation for this is f_0\simeq f_1.


The above two definitions are related, since a path is a special kind of map f: I\to X.

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