fundamental groups of open subsets of the plane This should be a very basic algebraic topology question.
The other day I was thinking about the fact that $P^2(R)$ has $\pi_1 = Z/2Z$.
On the other hand I thought to myself how something like this can never happen for, say, an open subset of the real plane $R^2$. It's a very intuitive fact but I can't prove it.
So I guess my general question is: can open subsets of $R^2$ have torsion elements in $\pi_1$?
Related to this: is there a classification of the homotopy types of open subsets of the plane?

The new question is very different from the original one. The answer to it is that every noncompact connected surface is homotopy-equivalent to a bouquet of circles, see proofs and references here. In particular, fundamental group is free and, hence, torsion-free. 
Thus, the answer to the last question is: Yes, there is a description, namely they all are disjoint unions of bouquets of circles. 
A (quite a bit) more difficult theorem is that the fundamental group of every open connected subset of $R^3$ is still torsion-free. In dimensions $\ge 4$ this is, of course, false. 
