Manifolds and Global Homeomorphism

Question

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.

Answer 1

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.

Answer 2

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.