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

thenonstandard reals, since there are different nonstandard ordered fields each with the transfer principle (indeed, of any desired cardinality). Under the continuum hypothesis, then there is a unique saturated model of size continuum, and so one gets a little categoricity this way, but in general, one must deal with the possiblility of several different fields all looking like nonstandard reals. $\endgroup$ – Joel David Hamkins Jan 25 '12 at 1:03a field of nonstandard reals.. but it is now not clear to me if the answer of my question may depend on the particular chosen field of nonstandard reals. $\endgroup$ – Valerio Capraro Jan 25 '12 at 7:20