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
Letand letbe a basis for
is continuous if and only if for eachis an open subset of
Supposeis continuous. Since eachis an open subset ofthen
Suppose for eachis an open set inLetbe an open subset of Sinceis a basis for
Eachis an open set henceis a union of open sets andis continuous.