# Equivalence of two definitions of a contractible space.

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We have defined space X to be contractible in two ways:

Definition 1: X is contractible if it is homotopy equivalent to a point; and

Definition 2: X is contractible if the identity map of X is null-homotopic.

Show that these two definitions are equivalent.

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##### Solution Summary

The equivalence of two definitions of a contractible space is proved.

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First, write down the definitions:

Two spaces, X and Y are homotopic, if there are two maps, f: X->Y and g:Y->X such the fg is homotopic to id_X and gf is homotopic to id_Y.

A map f is null-homotopic if it is homotopic to a constant map.

Suppose, ...

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