This is an exercise (4.48) from Apostol's book on analysis. S is an open connected set in $\displaystyle R^{n}$. Let T be a component of R^{n} - S. Prove that $\displaystyle R^{n} - T$ is connected.

I take it that the appropriate formulation of 'connected' for subsets of a topological space M is that $\displaystyle X \subset M$ is connected iff for any partition $\displaystyle X \subset A \cup B$, where A and B are nonempty disjoint sets open in M, either $\displaystyle X \subset A$ or $\displaystyle X \subset B$, and I know that a component of X is a maximally connected subset of X ie a subset $\displaystyle Y \subset X$ such that for all $\displaystyle Y \subset Z \subset X$, Z is connected iff $\displaystyle Z = Y$.

I can of course write down the algebraic formulation of the hypotheses of the problem, but don't see where to go with them. And I am sadly bereft of any geometric intuition to guide me.

Supplementary query: how do I get the LaTex symbols to work in the above? I am completely new to all this.