Proof That Any Connected Subset of a Set Consisting of Disjoint Open Subsets Must be Contained in One of Those Subsets
Theorem
Suppose
is a space such that
where
and
are nonempty, open subsets of
If
is any connected subset of
then either
or
Proof
Suppose
withand that
and
Then
are nonempty, disjoint, subsets of
open inand
Henceis not connected and either
or
hence eitheror