Proof That if Any Two Points in a Space are in a Connected Set, Then the Set is Connected

Theorem

Letbe a space such that for anya connected setexists such thatThenis connected.

Proof

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.