(path) connectness as a homotopy invariant
Theorem. Let and be arbitrary topological spaces with (path) connected. If there are maps and such that is homotopic to the identity map, then is (path) connected.
Proof: Let and be maps satisfying theorem’s assumption. Furthermore let be a decomposition of into (path) connected components. Since is (path) connected, then for some . Thus . Now let be the homotopy from to the identity map. Let be a path defined by the formula: . Since for all we have and is path connected, then . Therefore , but which implies that , so is (path) connected.
Straightforward application of this theorem is following:
Corollary. Let and be homotopy equivalent spaces. Then is (path) connected if and only if is (path) connected.
|Title||(path) connectness as a homotopy invariant|
|Date of creation||2013-03-22 18:02:15|
|Last modified on||2013-03-22 18:02:15|
|Last modified by||joking (16130)|