Tangent space of a point

Question

Let M = {p} be a manifold contained a single point p.

I'm wondering What is the tangent space of M at p ? And what is the vector space $\Omega(M)$ of smooth differential forms on M ?

Thank you!

Answer 1

A single point is $0$-dimensional, so the tangent space should also be a zero dimensional vector space. Similarly, you'll only have $\Omega^0(M)$ being nonzero ($\Omega^i(M)=0$ for $i>0$). These are just the (constant) functions as all functions on a single point are constant.