De Rham Cohomology

Hi!
I hope this is the right place to post this.
Also, I hope that TeX stuff works out as intended.

How do I compute the de Rham Cohomology Group H^1(\mathbb{R}^2 \backslash \{x\})?
And, for distinct x_1,...,x_n, what is H^1(\mathbb{R}^2 \backslash \{x_1,...,x_n\})?

After a bit of web research, I found out that the first one should be \mathbb{R} and the second \mathbb{R}^n, but I have no idea how to prove that.

Many thanks in advance!

i dunno lol

do a barrel role

The plane minus a point is smoothly homotopic to the circle, so you get R. For the general case I'd try induction and Mayer-Vietoris/excision. I think looking at elements could get pretty messy here, but if you want to do this you might want to start  by reading the last proofs in Spivak's chapter on De Rham cohomology.