Proof That a Function Continuous For Every Source Topology is Continuous With Respect to the Indiscrete Topology

Theorem

Letbe a family of topologies on a setand letbe continuous with respect to each topology

Thenis continuous with respect toSince the indiscrete topology is the intersection of every topology onis continuous with respect to the indiscrete topology.

Proof

The intersection of any topology is itself a topology.

Letbe any open subset ofso thatConsiderSinceis continuous with respect to each topologythe setbelongs to eachhence

Henceis continuous with respect to