0
$\begingroup$

I've recently been studying manifolds, and while I'm familiar with the technical definition, a topological space which is both Locally Euclidean and Hausdorff with perhaps a few other requirements depending on the author, I was wondering about the relation between a space being locally euclidean and globally euclidean.

When I say globally euclidean, I particularly mean when the entire space itself is homeomorphic to $\mathbb R^n$. I was wondering if there could be such a thing as a space which is globally euclidean but not locally euclidean. In other words, a space which is homeomorphic to $\mathbb R^n$ but which is not a manifold.

My intuition says no, but I'm not sure if this is one of those things where there are fringe examples that demonstrate the logic to be incomplete.

$\endgroup$
4
  • $\begingroup$ I don't know if it would be interesting for you but there are manifolds that are homeomorphic to $\mathbb{R}^n$ but not diffeomorphic. They are called exotic structures but they are of course still manifolds. $\endgroup$ – Levent Mar 9 '18 at 21:43
  • $\begingroup$ @Levent I've only just started studying manifolds, would it be wise to put that on hold and familiarize myself with Differential Geometry so that I can become better acquainted with the notion of diffeomorphisms? I'm already loosely familiar with them, but it's mostly a cursory knowledge, rather than a thorough understanding of their behavior. $\endgroup$ – Harrison Gourd Mar 9 '18 at 21:47
  • $\begingroup$ To be fair, my comment is actually completely irrelevant to the question. I just thought that it might be interesting for you. Apart from that, I believe that the theory of manifolds is the place where you learn about diffeomorphisms. $\endgroup$ – Levent Mar 9 '18 at 21:50
  • $\begingroup$ I've mostly been learning about them in the context of general topology, rather than Manifold theory itself. $\endgroup$ – Harrison Gourd Mar 9 '18 at 22:07
1
$\begingroup$

No, there is not. Because $\mathbb{R}^n$ is locally Euclidean. Therefore, any space which is homeomorphic to $\mathbb{R}^n$ will be locally Euclidean too.

$\endgroup$
0
$\begingroup$

Let $M$ be your space. If $\phi\colon M \to \Bbb R^n$ is a homeomorphism, for every $p \in M$ you can take the (topological) chart $(M, \phi)$ around $p$, so $M$ is locally Euclidean.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.