gn.general topology - A question about unbounded connected subsets of the plane.

The following gives a partial answer: no such unbounded connected set may exist with the further assumption that it is closed. Actually, the argument generalizes for any locally compact metric space. I'm not completely sure that a much simpler or even trivial proof may exist, though.

Let $Gamma$ be a closed unbounded connected subset of the plane. Let $xinGamma$ and let $B:=B(x,r)$ be an open ball around $x$. I claim that the connected component of $x$ in $Gammacap bar{B}$ meets $partial B$, which shows that $Gamma$ does contain non-trivial bounded connected subsets.

For any $epsilon>0$, consider the $epsilon$-neighborhood of $Gamma,$ that is $Gamma_epsilon:=cup_{yinGamma}B(y,epsilon).$ It is an open unbounded connected subset of the plane.
Let $U_epsilon$ be the connected component of $x$ in $Gamma_epsiloncap B$. Since the latter is locally connected, $U_epsilon$ is both an open and closed subset of it in the relative topology. It is therefore an open subset of $Gamma_epsilon$; however it is not closed in it, because $Gamma_epsilon$ is connected. Therefore $bar U_epsilon$ is a closed connected set that meets $partial B, $ and of course contains $x$. Since the set of all connected closed subsets of a compact metric space is compact in the Hausdorff distance, taking a limit as $epsilonto0$ we get a bounded connected subset of $Gamma$ connecting $x$ with $partial B$ (this also passes to the limit).

Rmk One could state the above in terms of the one-point compactification of $Gamma$, and more generally for compact connected metric spaces. The trick of approximating a metric space with a locally connected metric space is made possible via the Kuratowski embedding (one defines $X_epsilon$ as an $epsilon$ nbd of $X$ in the embedding).

PS: Of course the same affirmative conclusion holds, even more directely, if $Gamma$ is assumed to be open, which is another case included in the original assumption of completely metrizable.