Fundamental groups of path connected subspaces Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? 
For example, for convex subspaces we know that, but if we take only path connected subespaces what happen?
 A: Every reasonable path connected subspace of $\mathbb{R}^2$ has fundamental group a free group, which is either trivial or infinite. "Reasonable" means homotopy equivalent to a tubular neighborhood of it in $\mathbb{R}^2$, and in particular means homotopy equivalent to an open subset of $\mathbb{R}^2$. Now, an open subset of $\mathbb{R}^2$ is a noncompact surface, and every connected noncompact surface has fundamental group a free group; see, for example, this MO question. 
An example of an unreasonable path connected subspace of $\mathbb{R}^2$ is the Hawaiian earring, whose fundamental group is not free. 
A: Your question was answered in affirmative by George Lowther here:
Is the fundamental group of every subset of $\mathbb{R}^2$ torsion free?
More precisely, he proves that if $X$ is a subset of any closed surface $S$, for instance, $S=S^2$, and $x\in X$, then $\pi_1(X,x)$ is torsion free. 
Note that no assumption is made about the nature of the subset $X$, it is not assumed to be "reasonable". 
From this, it follows that if $\pi_1(X,x)$ is nontrivial, it contains a nontrivial element $\gamma$, which, then, necessarily has infinite order. Therefore, $\pi_1(X,x)$ is either trivial or infinite. 
