Denote $X \prec Y$ if $X$ can be embedded to $Y$. Give an example where $X \prec Y \prec X$, but $X$ is not homeomoprhic to $Y$.
$X=(0,1)\cup \{2\}, Y=(0,3)$. Note that $Y$ is connected but $X$ is not. $x\to \frac x 3$ is an embedding of $Y$ in $X$.
$X=(0,1)\cup \{2\}, Y=(0,3)$. Note that $Y$ is connected but $X$ is not. $x\to \frac x 3$ is an embedding of $Y$ in $X$.