Proof That the Union of Two Connected Sets Not Mutually Separated are Connected
Letandbe connected sets which are not mutually separated. Thenis connected.
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.
Thereforeis not a disconnection ofandis connected.