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 space
onto the set
with the discrete topology.
Proof
A spaceis connected if there are no nonempty, disjoint open sets
such that
Suppose
is continuous and onto.
and
are both open and closed in
so
are both open and closed in
But
and
since
is connected
