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