Details for SE(3) being a manifold

Question

As a student of engineering, i read that SE(3) is a manifold which commonly is known to us as a transformation matrix.

I have read proofs showing that a sphere is a 2-dimensional manifold. The proof has 3 components:-

1. the set
2. the topology assumed on the set
3. defining charts

For sphere the set is already given and the topology assumed is the standard one on $\mathbb{R}^2$. Thereafter they explicitly write down the 4 charts that maps this sphere to the plane.

This proof is very common so could be found easily. But i have not been able to find such an explicit proof characterizing these 3 ingredients of proving SE(3) as manifold. On which set are we talking about what topology and how this structure can be explicitly shown to be euclidean locally?

A detailed proof from scratch PLUS connection of this mathematical structure to the physical significance of transformation matrices in robotics would be appreciated. Or reference to that effect will also be helpful.