Explore BrainMass

Explore BrainMass

    Equivalence of two definitions of a contractible space.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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 June 4, 2020, 12:43 am ad1c9bdddf

    Solution Preview

    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, ...

    Solution Summary

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