Motivated by this [question](http://mathoverflow.net/questions/191120/the-letters-of-the-word-art)  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)