Motivated by this question we ask:

Up to homeomorphism, are there only a finite number of connected locally compact hausdorff topological space $X$ such that $X$ has an open set $U$ homeomorphic to $\mathbb{R}$ such that $X-U$ is homeomorphic to $\mathbb{R}$, too? (Note that there is only one disconnected possibility, so we add connectedness assumption)