Yeah, well, that's all the title bar allows me, but it's not quite what I meant.

So. I was thinking - just

*thinking*, mind - of extending my (arguably) crappy tutorial on topological spaces to some elementary manifold theory.

So if such a thread were to make any sense, I would have to be sure of the following, which I want to check on:

Considered as a vector space, it is certainly true that any proper (or improper) subspace of

is isomorphic to

itself.

Q: Is this the only vector space with this property? I think so, but I'm not certain.....

It is also certainly true that, considered as a topological space, any subspace of

is homeomorphic to

itself.

Q: Is this the only topological space with this property? I think so, but I'm not certain.....

Now consider

as a

manifold.

Q: Is it true that any submanifold of

is diffeomorphic to

itself? This seems less likely, but if so, would this be the only such manifold with this property?