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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.

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  • $\begingroup$ 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
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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.

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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.

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