## Proof That a Subspace is Connected if and Only if it Cannot Be Expressed as the Union of Nonempty, Mutually Separated Subsets

Theorem

A subsetofis connected if and only ifcannot be expressed as the unionof nonempty , mutually separated subsetsandof

Proof

Ifis not connected thenwhereandandare both open and closed in

Supposethen

Sincethenthen- a contradiction.

Similarly

Now supposeandthen

Similarly

Henceandare closed inandis disconnected.