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 ofwith the euclidean topology having more than one point are the intervals and the real numbers.

Proof

Supposeis connected and is not an interval. Then there existwith and

Thenandis a decomposition ofsince both sets are open, disjoint and nonempty.

Supposeis an interval and is not connected. Then disjoint, nonempty, open setsand exist such that

We can findwith

Defineand sinceis an interval,

We also haveand sinceis closed in

Butis also open inhenceexists such thatcontradiction the definition of