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?