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Edit: According to the Gelfand duality between topological space and commutative algebra, I add some new tags. So the question is that what is 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} $.

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

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.

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

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

Edit: According to the Gelfand duality between topological space and commutative algebra, I add some new tags. So the question is that what is 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} $.

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