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,
Valerio
Best Answer
As mentioned in the comments, the actual topology on the non-standard extension can be quite nasty. This is illustrated for example in the first set of problems in these notes. A solution is to replace standard topological notions by definable analogues. Then things mostly work in an arbitrary o-minimal structure. This is also explained in the above notes.
More specifically on algebraic topology in the o-minimal settings, there are several papers by Berarducci and by Edmundo