Question: Find two closed, connected subsets in $\mathbb{R}^2$, $A$ and $B$, such that $A$ is not homeomorphic to $B$, but there is a continuous bijection $f:A \rightarrow B$ and a continuous bijection $g:B \rightarrow A$.
This is a homework question, so please only very small hints. I realize that both $A$ and $B$ must not be compact. Since they both must be closed, then they must be unbounded. However, I am having a hard time getting started on this. It is very easy to find two closed, unbounded, connected subsets of the plane that are not homeomorphic to each other, but it is hard to find the continuous bijections required.
I know the classic example of a continuous bijection with a discontinuous inverse is a map $f: [0,2\pi) \rightarrow \mathbb{S}^1$ given by $f(x) = (\cos x, \sin x)$. I am trying to use this map as a template to come up with my sets but I am having no success.