Take the 2-minute tour ×  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 $\mat