Proof That a Subspace is Connected if and Only if it Cannot Be Expressed as the Union of Nonempty, Mutually Separated Subsets
Theorem
A subset
of
is connected if and only ifcannot be expressed as the union
of nonempty , mutually separated subsets
and
of
Proof
Ifis not connected then
where
andandare both open and closed in
Suppose
then
Since
then
then
- a contradiction.
Similarly
Now supposeand
then
Similarly
Henceandare closed inandis disconnected.