Proof That if Any Two Points in a Space are in a Connected Set, Then the Set is Connected
Letbe a space such that for anya connected setexists such thatThenis connected.
Letbe a fixed element ofthen for anythere exists a connected subspaceofsuch that
Consider the family of setsconsisting of all connected subsets of
The intersection of all theis nonempty since
Hence by this theorem is connected.