A topological space is the most general space that has a notion of continuity, but as we discovered in recent instalments of the Poincare Project, most applications of topological spaces are restricted to a subset known as Hausdorff Spaces.
We're actually going to go a step further now and define a structure called a manifold. A manifold is the most general space that can have a coordinate system. It is the generalisation of what is often referred to as a surface and the foundation for things like vector calculus.
Define a chart to be a continuous one-to-one mapping from an open set to R^n.
A manifold is a topological space covered by one or more charts. In other words, every point (and some open set the point is in) is part of at least one chart.
One way to think of this is that an n-dimensional manifold is locally (but not necessarily globally) like R^n. It is the fact that a sphere is a 2-dimensional manifold that allows us to draw flat 2-dimensional maps of sections of it.
A chart provides a coordinate system and the coordinates of a point on a manifold are just the components of the point in R^n that the point on the manifold maps to in that chart.
For some manifolds, it is possible to cover the entire space with a single chart. Others needs multiple charts. For example, no single coordinate system can cover a sphere continuously one-to-one; for example, the coordinate system of latitude and longitude breaks down at the poles where a single point on the sphere maps to an infinite number of points in R^2.
Much of the foundational work on manifolds is due to Poincaré himself.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.