a connected and locally path connected space is path connected
Proof. Let be the space and fix . Let be the set of all points in that can be joined to by a path. is nonempty so it is enough to show that is both closed and open.
To show that is closed: Let be in and choose an open path connected neighborhood of . Then . Choose . Then can be joined to by a path and can be joined to by a path, so by addition of paths, can be joined to by a path, that is, .
|Title||a connected and locally path connected space is path connected|
|Date of creation||2013-03-22 16:50:43|
|Last modified on||2013-03-22 16:50:43|
|Last modified by||Mathprof (13753)|