So far, we’ve been talking about topologies in the most general sense: point-set topology. As we’ve seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of open sets.
But most of the things that are commonly associated with topology aren’t just abstract point-sets: they’re *shapes* and *surfaces* – in topological terms, they’re things called *manifolds*.
Informally, a manifold is a set of points forming a surface that *appears to be* euclidean if you look at small sections. Manifolds include euclidean surfaces – like the standard topology on a plane; but they also include many non-euclidean surfaces, like the surface of a sphere or a torus.
The formal definition is very much like the informal, with a few additions. But before we get there, I’ll refresh your memories about some concepts that we’ll need.
- A set **S** is *countable* if/f there is a total, onto, one-to-one function *f : **S** → **Z**;*, (where **Z** is the set of natural numbers), mapping each element of **S** onto exactly one natural number.
- Given a topological space (**T**, τ), a *basis* β for τ is a collection of open sets from which any open set in τ can be generated by a finite sequence of unions and intersections of sets in β.
- A topological space (**T**, τ) is called a *Hausdorff* space if/f if for any two distinct points in T, each point is a member of *at least* one neighborhood that the other is not.
A topological space (**T**,τ) is an *n*-manifold if/f:
- τ has a *countable* basis.
- (**T**,τ) is a Hausdorff space.
- Every point in **T** has a neighborhood homeomorphic to an open euclidean *n*-ball.
Basically, what this really means is pretty much what I said in the informal definition. In a euclidean *n*-space, every point has a neighborhood which is shaped like an *n*-ball, and can be *separated* from any other point using an *n*-ball shaped neighborhood of the appropriate size. In a manifold, the neighborhoods around a point *look like* the euclidean neighborhoods.
If you think of a large enough torus, you can easily imagine that the smaller open 2-balls (disks) around a particular point will look very much like flat disks. In fact, as the torus gets larger, they’ll become virtually indistinguishable from flat euclidean disks. But as you move away from the individual point, and look at the properties of the entire surface, you see that the euclidean properties fail.
Another interesting way of thinking about manifolds is in terms of *charts*, and charts will end up being important later. A *chart* for an manifold is an invertable map from some euclidean manifold to *part of* the manifold which preserves the topological structure. If a manifold isn’t euclidean, then there isn’t a single chart for the entire manifold. But we can find a *set* of *overlapping* charts so that every point in the manifold is part of *at least* one chart, and the edges of all of the charts overlap. A set of overlapping charts like that is called an *atlas* for the manifold, and we will sometimes say that the atlas *defines* the manifold. For any given manifold, there are many different atlases that can define it. The union of all possible atlases for a manifold, which is the set of *all* charts that can be mapped onto parts of the manifold is called the *maximal* atlas for the manifold. The maximal atlas for a manifold is, obviously, unique.
For some manifolds, we can define an atlas consisting of charts *with coordinate systems*. If we can do that, then we have something wonderful: a topology on which we can do angles, distances, and most importantly, *calculus*.
Topologists draw a lot of distinctions between different kinds of manifolds; a few interesting examples are:
- A *Lie group* is a manifold with a valid closed *product* operator between points in the manifold.
- A *Reimann* manifold is a manifold on which you can meaningfully defined angles and distance.
- A *differentiable* manifold is one on which you can do calculus.