I am searching for a two way embedding between two non-homeomorphic spaces. In other words, I want two non-homeomorphic spaces such that $X$ is embedded in $Y$ and $Y$ is embedded in $X$.
- Recall given two spaces $(X, \mathfrak{T})$ and $(Y, \mathfrak{J}), A \subseteq Y$, if there is a homeomorphism $f: X \to A$ then we call $X$ embedded in $Y$
My initial attempt is to create a space with copies in each set like:
Let $X = (0,1) \cup \{2\}$, and $Y = (0,1)$
The two spaces are obviously not homeomorphic. Then take the identitify function $id(Y) = A \subset X$, then $Y$ is embedded in $X$. But this way $X$ is not embedded in $Y$, since we cannot map $\{2\}$ into $Y$ through a bijective function.
What modification can I make to the spaces to get the two way embedding without them being homeomorphic.
You can consider $[0,1]$ and $\mathbb{R}$.