Proof That a Continuous, Open Function From a Locally Compact Space Onto a Space Produces A Locally Compact Space
Ifis a continuous,, open function from a locally compact spaceonto a spacethenis also locally compact.
A functionis said to be open if, for any open subsetis open in
Letand letbe a neighbourhood ofFor some
Sinceis continuous, an open setexists such thatand
Sinceis locally compact, there is a compact setsuch that
Sinceis open,is open. Sinceis compact andis continuous,is compact. Henceis locally compact.