Theorem

Suppose

Thenis continuous if and only ifthere exists an open setwith

The converse is also true.

Proof

Supposeis continuous. Take anyand any open neighbourhoodof

Thenand

Sinceis continuousSincewe have

Letrepresent an open set ofSupposethensois an open neighbourhood ofAn open setexists such thatand

Buthence for eachwhere

Henceis the union of open subsets ofand so is an open subset.