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

Add comment

Security code
Refresh