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?
 A: 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.
A: I will use the fact that if $X$ is a space, $x$ a point, and $A$ and $B$ two connected subspaces of $X$ both containing $x$, then $A\cup B$ is connected. (This holds more generally for any arbitrary union.)
Fix some point $x$ in $S$, and for each element $y$ of $S$, let $S_y$ denote a connected subspace of $S$ containing $x$ and $y$.  Then $S=\bigcup_{y\in S}{S_y}$ is connected as it is a union of connected subspaces all containing a common point ($x$).
A: Suppose S is a set, any pair of points of S (P,Q assume) can be contained in a connected subset of S. 
I think you can still use the polygonal chain theorem, but I think you've slightly misquoted it:  
Polygonal Chain Theorem: Let G be an open set.  Then G is connected if and only if every pair of points of G can be connected by a polygonal chain in G. (courtesy of Michael Henle's Combinatorial Introduction to Topology)
If each for each arbitrary pair $P_1$, $P_2$  in S is contained in a connected subset $G$ of S, then (by the Polygonal Chain Theorem) every pair of points in $G$ can be connected by a polygonal chain in $G$.  Then there exists a polygonal chain connecting $P_1$ and $P_2$, and hence a polygonal chain exists connecting any pair of points in S.  Hence, by the Polygonal Chain Theorem, S is connected.
