Proving connectedness Suppose $S$ is a set, any pair of points of $S$ ($P,Q$ assume) can be contained in a connected subset of $S$.  Show $S$ is also connected.
I tried to use the polygonal chain theorem(Every open set can be connected by a polygonal chain.) but we don't know if $S$ is an open set.
Using the original definition of connectedness seems really abstract...
any help?

Picture: If $S$ had two separate pieces, pick a point in each piece. Put these two points in a connected subset (using the assumption), and conclude they really weren't in separate pieces.
Formalization: Suppose that $S = U \cup V$ with $U$ and $V$ open, disjoint, non-empty subsets of $S$. Choose $p \in U$, $q \in V$ and $A \subseteq S$ containing $p, q$ with $A$ connected. Consider the sets $A \cap U$ and $A \cap V$, and try to derive a contradiction.
