Proof That Continuous Open Functions Preserve Local Connectedness
Ifis a continuous open function from a locally connected spaceonto a spacethenis locally connected.
Supposeis continuous, open and onto
Letandbe any neighbourhood of
Sinceis onto,exists such that
is continuous, sois open inandis locally compact, so there is a connected neighbourhoodofsuch that
Sinceis continuous and open,is a connected open set andhenceis locally connected.