Proof That Any Connected Subset of a Set Consisting of Disjoint Open Subsets Must be Contained in One of Those Subsets

Theorem

Supposeis a space such thatwhereandare nonempty, open subsets of

Ifis any connected subset ofthen eitheror

Proof

Supposewithand thatand

Thenare nonempty, disjoint, subsets ofopen inand

Henceis not connected and eitherorhence eitheror