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

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.

• Start thinking about dimensions 1 and 2. – Anubhav Mukherjee Oct 19 '17 at 22:24