The Dimension of a Manifold
One thing that may seem clear at first blush about manifolds actually takes a little thought to be sure about. Specifically, our definition says that each point is contained in some coordinate patch , which comes with a coordinate map . But does the here have to be the same for all points ?
Well, let’s start by considering what happens if two coordinate patches intersect each other. Let and be open subsets with not empty, and with coordinate maps and . Now we can restrict our coordinate patches to the intersection, getting and . These are two different coordinate patches with two different coordinate maps on the same open subset .
So what? So now we can use in reverse to lift up from into , and then use forward to drop down into . This composition is thus a homeomorphism from an open region in to an open region in . And this is absolutely impossible unless .
So now we know that any two coordinate patches that intersect must use the same value of . Does this mean that we always have to use the same value of ? Well, not quite.
Take two distinct natural numbers and . For each one, we can come up with all the coordinate patches with that dimension, and take their unions and . Since the union of any collection of open sets is open, each of these sets must be open. But they can’t intersect, or else some coordinate patch with dimension and some other with dimension would have to intersect, and we just saw that they can’t.
The only way this is possible is for and to live in different connected components of . And, indeed, our definition so far doesn’t rule out this possibility. We could have a two-dimensional sphere and a one-dimensional circle floating next to each other, never touching, and they would count as a manifold according to what we’ve said so far.
There are two ways around this. One is to only ever talk about connected manifolds, which automatically have a unique dimension since they only have one connected component. However, this imposes restrictions, like making it difficult to take intersections of manifolds and have the result still be a manifold, since it may suddenly be disconnected. The other alternative, which we will use, is simply to assert that all connected components of a manifold must have the same dimension.
Either way, we can specify this dimension by saying that is an -dimensional manifold, or an -manifold for short.