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> **Edit:** According to the Gelfand duality between topological spaces and commutative $C^{\*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $C\_{0}(\mathbb{R})$. One can think to stabilization of this question, that is $A=C\_{0}(\mathbb{R})\otimes \mathcal{K} $ where $\mathcal{K}$ is the algebra of compact operators.
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The following was the first version of my question:


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 $\mathbb{R}^{2}$?



The letters of the word "ART"
Yes there are infinitely many, by a version of Mike Jury's idea. In fact there are uncountably many embeddable in $\mathbb R^2$. Take the union of two real curves:


the open one $y= x^{-1}\sin(1/x)$


the closed one $x= f(y)$ for $f$ a function that is $0$ on some closed set $S$ and negative elsewhere. 


The closure of the open curve is exactly $S$, so we can identify $S$ from the topology of the space. Clearly there are uncountably many closed subsets of $\mathbb R$ up to homeomorphism of $\mathbb R$ (e.g. encode a real number as a sequence of closed intervals, isolated points, and Cantor sets.) So there are uncountably many spaces.


