product of path connected spaces is path connected
Proof. ”” Assume that and are path connected and let be arbitrary points. Since is path connected, then there exists a continous map
Analogously there exists a continous map
Then we have an induced map
defined by the formula:
which is continous path from to .
”” On the other hand assume that is path connected. Let and . Then there exists a path
We also have the projection map such that . Thus we have a map
defined by the formula
This is a continous path from to , therefore is path connected. Analogously is path connected.
|Title||product of path connected spaces is path connected|
|Date of creation||2013-03-22 18:31:02|
|Last modified on||2013-03-22 18:31:02|
|Last modified by||joking (16130)|