Proof That a Function is Continuous if and Only if the Inverse Image of Every Basis Element of the Image Topology is Open in the Source Topology

Theorem

Letand letbe a basis for

is continuous if and only if for eachis an open subset of

Proof

Supposeis continuous. Since eachis an open subset ofthen

Suppose for eachis an open set inLetbe an open subset of Sinceis a basis for

Then

Eachis an open set henceis a union of open sets andis continuous.