Are there only a finite number of connected topological spaces $X$ (up to homeomorphism) with the property that $X$ has an open subset $U$ such that $U$ and $X-U$ are homeomorphic to $\mathbb{R}$? I know three examples as I wrote in the title of this question. (We delete the end critical points from each letter.) Among capital alphabet, there are no other topological type with the above property.

Is it true that any space $X$ with this property can be embedded in $\mathbb{R}^{2}$?