Non standard Algebraic Topology - MathOverflow

Non standard Algebraic Topology

2012-01-24T19:15:21Z

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

Answer by Tim Porter for Non standard Algebraic Topology

2012-01-24T21:11:41Z

Qu 2. In the 1970s there were some papers dealing with the non-standard analysis and the theory of shape in the sense of Borsuk. The author was Frank Wattenberg. The reference is Fund. Math. 98 (1978), 41-60.

I do not know if more was published although I did see a preprint of another paper. I also do not know if those ideas have been followed up.

Answer by Moshe for Non standard Algebraic Topology

2012-01-24T22:40:58Z

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