The central object of study in differential topology and differential geometry is a “manifold”. This is a topological space, which looks “close-up” like a real vector space. In fancier language, a manifold is “locally homeomorphic” to for some .
We know what a homeomorphism is: it’s the right notion of isomorphism in the category of topological spaces and continuous maps. More specifically, it’s a bijective function — so it has an inverse — that is continuous in both directions. Topological spaces connected by a homeomorphism are effectively “the same”.
Now, we aren’t asking that a manifold be homeomorphic to -dimensional real space as a topological space. That would be far too restrictive, and we’d just get those vector spaces again, without the vector space structure of course. Instead, we want our spaces to be locally homeomorphic. That means that each point should be contained in some neighborhood — an open set — and that this neighborhood should be homeomorphic to a neighborhood in for some .
We should note, here, that a manifold doesn’t require you to come up with explicit neighborhoods and homeomorphisms for every point. In fact, for any point there are usually many neighborhoods and many homeomorphisms that make locally homeomorphic to at , and any one of them is just as good as any other.
We should also note that many sources insist that the neighborhood be homeomorphic to all of . It turns out that this is an equivalent condition, but the connection is really more trouble than it’s worth. Given that it’s a lot easier to give explicit examples if we don’t restrict ourselves like this, we’ll just leave it out. However, to simplify language, we will often just say that a point has a neighborhood “homeomorphic to ” instead of “homeomorphic to an open set in “.
As a rough example (which will be made more explicit in the future), consider the two-dimensional sphere . This is the collection of all the points with , with the subspace topology from . And it’s pretty well-modeled by the surface of the Earth!
So think of yourself, on the surface of the Earth (as I’m pretty sure most of you are). As you look around yourself, you can see a few miles in any direction — barring local obstacles like buildings or mountains. This gives a neighborhood, and in this neighborhood the Earth looks, well, pretty much like a flat plane! That is, locally — within a few miles of any given point — the sphere of the Earth’s surface looks like the plane . Of course we know that as we zoom out the Earth is not topologically a plane, but close up it looks like it is.
As a technical point here, the important point isn’t that the Earth looks flat, but that it looks like a plane. Currently we’re just talking about differential topology, which only concerns itself with general shapes and not with measurements of things like curvature. Even if you include bumpy features like mountains and valleys, the Earth’s surface is locally homeomorphic to a plane and that’s all that matters now.
To be a little more explicit with this sphere example, imagine a rubber ball; the surface is again a topological sphere. Now, cut a hole in the ball and stretch the hole open. If we imagine that the material of the ball is infinitely stretchy, we can pull it out until it forms a big flat disk that we can lay in the plane. Thus, for every point except the ones in that hole we’ve shown a homeomorphism from an open set containing the point to an open set in the plane. We can go back and cut a hole in a different area of the ball and do the same thing, which shows that even the points inside our original hole have a neighborhood homeomorphic to . As a side benefit, we’ve seen that most of the points on the ball have two homeomorphisms (at least!), which is perfectly fine. The definition of a manifold only asks that there be at least one.