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.