Proof That if Any Two Points in a Space are in a Connected Set, Then the Set is Connected
Theorem
Let
be a space such that for any
a connected set
exists such that
Thenis connected.
Proof
Let
be a fixed element of
then for any
there exists a connected subspace
ofsuch that
Consider the family of sets
consisting of all connected subsets of
The intersection of all theis nonempty since
Hence by this theorem is connected.
