Proof That Continuous Open Functions Preserve Local Connectedness

Theorem

Ifis a continuous open function from a locally connected spaceonto a spacethenis locally connected.

Proof

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.