Proof That There is No Continuous Function From a Connected Space Onto The Set {0,1} With the Discrete Topology

Theorem

There is no continuous function from a connected spaceonto the setwith the discrete topology.

Proof

A spaceis connected if there are no nonempty, disjoint open setssuch that

Supposeis continuous and onto.

andare both open and closed insoare both open and closed in

Butandsinceis connected