Explore BrainMass
Share

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 March 21, 2019, 8:58 pm ad1c9bdddf
https://brainmass.com/math/linear-algebra/equivalence-of-two-definitions-of-a-contractible-space-355942

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.

$2.19