As promised. I’m back to a conceptual description. What I think I’ll do is look to the post before and try to tease apart a single term into its concept.
What is a smooth manifold? Well first of all there are two parts to this “smooth” and “manifold.” We’ll start with manifold, and I’ll quickly say that smooth is a technical term so probably doesn’t mean quite what you think.
A manifold is a topological idea. It is also a “local” idea. This means that different manifolds can look radically different, but when you zoom in at a very small scale they start to look the same. Basically, manifolds are an attempt to generalize are what we generally think of as flat space. The surface of your table or a piece of graph paper gives you 2-dimensional space. It is flat and you can put nice perfect squares on it to give you a location. Likewise you can chunk out cubes of the air around you to give you three dimensional coordinates.
When we generalize we can have curved weird looking things, as long as at any given place we can zoom in really close and get something that looks like a piece of our flat nice space. This is the property of being “locally Euclidean.” Try it with your coffee cup. Zoom in really close at any chosen place. Pretend you’re an ant crawling around on the cup. If you were a small enough ant, even the curved parts are going to seem flat to you. So examples of a 1-dimensional would be a circle or curve. Examples of 2-dimensional manifolds would be the surface of your coffee cup or the surface of a sphere. Examples of 3-dimensional manifolds would be … heh. Notice how 1-dimensional manifolds have been living in 2 dimensional space and 2 dimensional manifolds have been living in 3-dimensional space. (There is a theorem about this I won’t get to that any n-dimensional manifold can live in 2n-dimensional space.)
There are technically two other properties that we’ve ignored. A manifold has to be second countable and Hausdorff. This was a triviality since the examples I gave were living in (were a subspace of) a higher dimensional space that we knew had these properties. So second countable is a pretty abstract idea. Technically it means that there is a countable basis. Or if you take the numbers 1,2,3,… you can for each number assign an open set. Taking this collection we can get all the other open sets. This is sort of meaningless since I didn’t describe a topological space… Hausdorff is easier. It means that given any two points that aren’t the same, we can zoom in close enough on each one, so that the zoom in of the one doesn’t have any of the same points as the zoom in of the other.
So you’re thinking, all those properties seem to be everywhere, how could something not satisfy those? That’s because a manifold is a generalization of our intuition of space. All the concepts of it are meant to be intuitive. It actually takes a little bit of work to find things that don’t satisfy those properties. I think those examples are for another time (along with what it means to be a smooth manifold).