Showing equivalence between the Jordan Curve Theorem and another similar theorem.

Question

In Rolfsen's Knots and Links there is an exercise to show the equivlance between the the JCT and another theorem:

1. JCT: If $J$ is a simple closed curve in $R^2$, then $R^2-J$ has two components, and $J$ is the boundary of each.

2. TH: If $L$ is a closed subset of $R^2$ which is homeomorphic with $R^1$, then $R^2-L$ has two components, and $L$ is the boundary of each.

I understand that $L$ can't be bounded, else it will be compact and not homeomorphic to $R^1$. Also $L$ is not homeomorphic to $J$ since removing a point from $R^1$ leaves it disconnected but not $J$. Kindly provide hints on how to proceed.