Topology given to the tangent space of a manifold at a given point.

Question

I'm learning differential topology from Lee's Introduction to Manifolds and was reading about the tangent space $T_pM$ of a smooth manifold $M$, of dimension say, $n$ at some point $p \in M$. Now, I do understand that $T_pM$ is an $n$ dimensional vector space, but I'm having trouble understanding what does an open set in $T_pM$ look like. I can't visualize what it is. Any help towards that end is appreciated.

Answer 1

Since $T_pM$ is finite dimensional, it is isomorphic to $\mathbb{R}^n$ for some $n$. It is endowed with the standard topology.

Remark: Of course the isomorphism $T_pM\cong\mathbb{R}^n$ is not canonical, but any such isomorphism gives the same topology, so it doesn't really matter in this situation.