Does there exist two topological space $X$ and $Y$ such that $X$ and $Y$ imbed in each other, $X$ is a quotient of $Y$, $Y$ is quotient of $X$, but $X$ and $Y$ are not homeomorphic?
The spaces $[0,1]$ and $(0,1)$ imbed in each other and are not homeomorphic. Also, $[0,1]$ is a quotient of $(0,1)$, but $(0,1)$ is not a quotient of $[0,1]$.
The spaces $[0,1]$ and $S^1$ are quotients of each other and are not homeomorphic. Also, $[0,1]$ imbeds in $S^1$, but $S^1$ does not imbed in $[0,1]$.
I'm wondering if there are two topological spaces which satisfy these properties. I cannot find a pair of examples, but I can't prove the impossibility without some nontrivial relationship between the imbeddings/quotients.
Any help in proving the impossibility or demonstrating a pair of examples is appreciated.