## Proof That the Union of Two Connected Sets Not Mutually Separated are Connected

Theorem

Letandbe connected sets which are not mutually separated. Thenis connected.

Proof

In a topological spaceare said to be mutually separated ifand

Supposeis disconnected then there exist disjoint nonempty open setsandsuch thatwithor

Ifandthenandare separated sets - a contradiction.

Thus eitheror

Thereforeis not a disconnection ofandis connected.