Non standard Algebraic Topology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:56:56Z http://mathoverflow.net/feeds/question/86562 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86562/non-standard-algebraic-topology Non standard Algebraic Topology Valerio Capraro 2012-01-24T19:15:21Z 2012-01-25T19:05:54Z <p>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.</p> <blockquote> <p><strong>Question:</strong> Does there exist an homeomorphism between *$\mathbb R^3$ and *$\mathbb R^2$?</p> </blockquote> <p>Well, this is the simplest subquestion of the most general one</p> <blockquote> <p><strong>Question:</strong> Is there anybody developing non standard Algebraic Topology? If not, is there any particular reason?</p> </blockquote> <p>Thanks in advance,</p> <p>Valerio</p> http://mathoverflow.net/questions/86562/non-standard-algebraic-topology/86569#86569 Answer by Tim Porter for Non standard Algebraic Topology Tim Porter 2012-01-24T21:11:41Z 2012-01-24T21:11:41Z <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/86562/non-standard-algebraic-topology/86582#86582 Answer by Moshe for Non standard Algebraic Topology Moshe 2012-01-24T22:40:58Z 2012-01-24T22:40:58Z <p>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 <a href="http://math.haifa.ac.il/kobi/o-minimal.pdf" rel="nofollow">these notes</a>. A solution is to replace standard topological notions by <em>definable</em> analogues. Then things mostly work in an arbitrary o-minimal structure. This is also explained in the above notes. </p> <p>More specifically on algebraic topology in the o-minimal settings, there are several papers by Berarducci and by Edmundo</p>