Surface where number of coordinate charts in atlas has to be infinite
In the definition of a parametrised surface $S$, for every point in the surface, $p \in W \subseteq S$, where $W$ is open, there exists a coordinate chart or patch , $F :U\to \mathbb{R}^n$ that maps to $p$ from an open subset $U \in \mathbb{R}^n$
Is that right? If anyone knows of a more general definition, I'm willing to learn. It sounds a lot like a manifold, which I'm not entirely familiar with.
In this definition, the number of surface patches in the atlas is not stipulated. Given a parametrisable surface, is a finite number of charts sufficient to describe the surface? Can we find a surface that requires infinitely many patches to fully chart? If so, in which $\mathbb{R}^n$ does the first such surface occur? In which dimensions is it always possible to find a finite number of patches for any given surface?
EDIT: Added requirement that such a surface (manifold) be connected. One made from infinitely many disconnected subsets would have to be charted infinitely.
If $M$ is an $n$-dimensional manifold, then it is in particular a normal space of covering dimension $n$, and Ostrand's theorem tells us that that if we start with a locally finite open covering $\mathcal U$ of $M$ consisting of coordinate charts (such a thing exists), there is a refinement $\mathcal V$ of $U$ such that $\mathcal V$ is the union of $n+1$ subfamilies $\mathcal V_1$, $\dots$, $\mathcal V_{n+1}$ such that each $\mathcal V_i$ is a disjoint family.
If we allow for non-connected coordinate charts, then by looking at the union of each $\mathcal V_i$ we obtain an atlas consisting of $n+1$ charts. (If we insist on connected charts, we should be able to connect the components with thing open tubes...)
This gives Ryan's bound for $n=3$, for example.
Later. One knows that the clique number of a graph is a lower bound for the chromatic number: Ostrand's theorem says that up to refinement, equality can be achieved if the graph comes from a covering of a normal space. This is a very nice result!
I think for a compact connected surface (a 2-dimensional manifolds) you only need three charts to cover the manifold. The rough idea is like this: think of a minimal CW-decomposition of the surface. Your first chart is a regular neighbourhood of the maximal forest in the 1-skeleton. The 2nd chart will be the union of the regular neighbourhoods of the remaining 1-cells, plus little arcs that connect them all up. Your remaining chart will be the union of the interiors of the 2-cells, with little arcs connecting them together.
Similarly, I think all compact 3-manifolds can be covered by 4 or 5 charts.
You can get lower bounds on the number of charts needed for any manifold by cup product arguments -- see: http://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category