Proof That a Subspace is Connected if and Only if it Cannot Be Expressed as the Union of Nonempty, Mutually Separated Subsets
A subsetofis connected if and only ifcannot be expressed as the unionof nonempty , mutually separated subsetsandof
Ifis not connected thenwhereandandare both open and closed in
Sincethenthen- a contradiction.
Henceandare closed inandis disconnected.