Counterexemple to Urysohn's lemma in a topos without denombrable choice ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:32:54Z http://mathoverflow.net/feeds/question/95257 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95257/counterexemple-to-urysohns-lemma-in-a-topos-without-denombrable-choice Counterexemple to Urysohn's lemma in a topos without denombrable choice ? Simon Henry 2012-04-26T14:03:13Z 2012-04-26T15:38:36Z <p>Hello !</p> <p>The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the law of excluded middle).</p> <p>I would like to find a counterexample to this theorem in the internal logic of a topos in which the axiom of countable choice does not hold (for exemple, the topos of smooth action of some non discrete locally pro-finite group, or the topos of sheaf on [0,1].)</p> <p>I need a counterexample which is compact, but If you have an example involving not a topological space but a local (an example of compact regular local which does not have enough functions with value in the Dedekind real) it's perfectly fine for me.</p> <p>Thank you !</p> http://mathoverflow.net/questions/95257/counterexemple-to-urysohns-lemma-in-a-topos-without-denombrable-choice/95262#95262 Answer by Andreas Blass for Counterexemple to Urysohn's lemma in a topos without denombrable choice ? Andreas Blass 2012-04-26T15:18:42Z 2012-04-26T15:18:42Z <p>Urysohn's Lemma is not provable in ZF (without the axiom of choice but with classical logic), so a suitable model of ZF will provide a topos of the sort you want. Checking the standard reference for such questions, "Consequences of the Axiom of Choice" by Paul Howard and Jean Rubin, I find the following permutation model (of ZF with atoms), due to Läuchli, in which Urysohn's Lemma is false. Begin with a countable set of atoms ordered like the rationals, take the group of all order-automorphisms, and take as supports the sets \$E\$ of atoms such that \$E\$ has only finitely many accumulation points and every infinite subset of \$E\$ has an accumulation point. </p> <p>A permutation model of ZF with atoms suffices to give a topos of the sort you want, but if you'd rather have a model of full ZF (i.e, without atoms), the Jech-Sochor embedding theorem lets you eliminate the atoms from Läuchli's example.</p>