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.© BrainMass Inc. brainmass.com March 21, 2019, 8:58 pm ad1c9bdddf
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.
The equivalence of two definitions of a contractible space is proved.