This is an exercise (4.48) from Apostol's book on analysis. S is an open connected set in

. Let T be a component of R^{n} - S. Prove that

is connected.

I take it that the appropriate formulation of 'connected' for subsets of a topological space M is that

is connected iff for any partition

, where A and B are nonempty disjoint sets open in M, either

or

, and I know that a component of X is a maximally connected subset of X ie a subset

such that for all

, Z is connected iff

.

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.