Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the $\ell^1$-(generalized)-metric.
Question: Does there exist an homeomorphism between *$\mathbb R^3$ and *$\mathbb R^2$?
Well, this is the simplest subquestion of the most general one
Question: Is there anybody developing non standard Algebraic Topology? If not, is there any particular reason?
Thanks in advance,