Eek! None of these drafts went up on time!
In principle, we know what a submanifold should be: a subobject in the category of smooth manifolds. That is, a submanifold of a manifold should be another manifold, along with an “inclusion” map which is smooth and left-cancellable.
On the underlying topological space, we understand subspaces; first and foremost, a submanifold needs to be a subspace. And one easy way to come up with a submanifold is just to take an open subspace. I say that any open subspace is automatically a submanifold. Indeed, if is a coordinate patch on , then is a coordinate patch on . The intersection is an open subset, and the restriction of to this intersection is still a local homeomorphism. Since the collection of all coordinate patches in our atlas cover all of , they surely cover as well.
As a quick example, an open interval in the real line is automatically an open manifold of , and so it’s a manifold. Any open set in any -dimensional real vector space is also automatically an -manifold.
More generally, it turns out that what we want to consider as a “submanifold” is actually somewhat more complicated, and we will have to come back to this point later.