(path) connectness as a homotopy invariant

Theorem. Let X and Y be arbitrary topological spacesMathworldPlanetmath with Y (path) connectedPlanetmathPlanetmath. If there are maps f:XY and g:YX such that gf:XX is homotopicMathworldPlanetmath to the identity map, then X is (path) connected.

Proof: Let f:XY and g:YX be maps satisfying theorem’s assumption. Furthermore let X=Xi be a decomposition of X into (path) connected componentsMathworldPlanetmathPlanetmath. Since Y is (path) connected, then g(Y)Xi for some i. Thus (gf)(X)Xi. Now let H:I×XX be the homotopyMathworldPlanetmath from gf to the identity map. Let αx:IX be a path defined by the formula: αx(t)=H(t,x). Since for all xX we have αx(0)Xi and I is path connected, then αx(I)Xi. Therefore H(I×X)Xi, but H({1}×X)=X which implies that Xi=X, so X is (path) connected.

Straightforward application of this theorem is following:

Corollary. Let X and Y be homotopy equivalent spaces. Then X is (path) connected if and only if Y is (path) connected.

Title (path) connectness as a homotopy invariant
Canonical name pathConnectnessAsAHomotopyInvariant
Date of creation 2013-03-22 18:02:15
Last modified on 2013-03-22 18:02:15
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 55P10
Related topic Homotopy
Related topic homotopyequivalence
Related topic path
Related topic connectedspace