Mutually densely-embedded non-homeomorphic topological spaces

Mutually densely-embedded non-homeomorphic topological spaces

My question is the following:
Do there exist two non-homeomorphic topological spaces $X$ and $Y$ such
that there are embeddings $f : X \hookrightarrow Y$, $g : Y
\hookrightarrow X$, with both $f(X)$ dense in $Y$ and $g(Y)$ dense in $X$?
It is easy to see that the answer is positive if one drops the requirement
that the image be dense for one of the embeddings. For example, let $X =
[0, 1[$, $Y = [0, 1]$, $f : X \rightarrow Y$ the inclusion, $g : Y
\rightarrow X$ defined by $g(y) = \frac{1}{2} y$ for any $y \in Y$. It is
clear that $f$ and $g$ are embeddings, and $f(X)$ is dense in $Y$, but
$g(Y)$ is not dense in $X$.
However, I can't figure out an example of the stronger situation above,
and I wouldn't know how to prove that in such a case the two spaces must
in fact be homeomorphic.
Thanks in advance.