Proof That The Only Connected Subsets of the Real Numbers With the Euclidean Topology Having More Than One Point are the Intervals and the Real Numbers
Theorem
The only connected subsets of
with the euclidean topology having more than one point are the intervals and the real numbers.
Proof
Suppose
is connected and is not an interval. Then there exist
with
and
Then
and
is a decomposition ofsince both sets are open, disjoint and nonempty.
Supposeis an interval and is not connected. Then disjoint, nonempty, open sets
and
exist such that
We can find
with
Define
and sinceis an interval,
We also have
and since
is closed in
Butis also open inhence
exists such that
contradiction the definition of