I was reading the definition of smooth manifold and i am little bit of confused. Informally it says
A smooth manifold is a topological manifold (i.e. a topological space locally homeomorphic to a Euclidean space) equipped with an equivalence class of atlases whose transition maps are all smooth. Here transition maps are from Euclidean space to Euclidean space.
Now what if I replace the word Euclidean space by topological vector space $\mathbb{R}^n$. Still we will get some object as we can talk about smoothness in topological vector space $\mathbb{R}^n$ without using any reference of coordinate system. So how much difference is there if I do the replacement? And my other question is
What are the properties of Euclidean space, we use to study Smooth manifolds, which are not present in $\mathbb{R}^n$ just as a topological vector space.
Till now what I have found that to describe local coordinates in manifold or to describe basis in tangent space one need a coordinate system in $\mathbb{R}^n$ so that you can pull it back to the manifold to define local coordinate there. Apart from these, where do we use the properties of Euclidean space to study smooth manifold?
Best Answer
Euclidean spaces are, by definition, $\mathbb{R}^{n}$. See if this clears things up for you. If not, come back and I'll try to help you some more.
Also, as the previous comment noted, any open set homeomorphic to an open ball will also be homeomorphic to all of $\mathbb{R}^{n}$ since open balls are homeomorphic to $\mathbb{R}^{n}$. Hence, in the definition of locally Euclidean, it does not matter if we a priori decided that our spaces should be locally homeomorphic to all of $\mathbb{R}^{n}$, an open ball, or an open set.