I'm on page 20 of 1215 of Gravitation: I couldn't help glancing into this before heading to the office this morning. 20p later I reluctantly dragged myself away. It's already provided a neat insight into co-ordinate systems.
People talk about needing multiple co-ordinate "patches" to cover a manifold. Why can't you just use one? Sometimes you can, e.g. a flat piece of paper. But what about a sphere? You can't wrap a flat piece of paper round a sphere (which is why map-making is such a pain). Any co-ordinate system you use on a sphere has a problem at two points (the poles) where all the lines of latitude meet and everything goes to hell in a hand basket ("Eveyerything goes to hell in a handbasket" is the practical definition of a singularity.) How do you get round this? Define TWO co-ordinate systems which don't have their poles overlapping. Now if you are at a singularity in one co-ordinate system, you just use the other system instead. You can do this for any co-ordinate singularity you find on a manifold. The system need only apply to a patch of the manifold, not necessarily the whole thing like with a sphere - hence "co-ordinate patches." You can invent as many as you need to cover all the singularities.
Topology joke my brother used to tell:
Q: How can you escape any prison cell, only using mathematics?
A: Simple! perform a co-ordinate transform such that the outside of the cell becomes the inside and vice versa - you are now free!