contractible open sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:22:21Z http://mathoverflow.net/feeds/question/100612 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100612/contractible-open-sets contractible open sets Juan Ahtziri 2012-06-25T17:53:44Z 2013-01-10T23:22:00Z <p>If $U\subset\mathbb{R}^n$ is an open and contractible subset such that there is a continuous function $f\colon U\to\mathbb{R}$ with only one minimum and the level curves of $f$ are connected by paths, then is $U$ homeomorphic to $\mathbb{R}^n$?</p> http://mathoverflow.net/questions/100612/contractible-open-sets/100629#100629 Answer by Juan Ahtziri for contractible open sets Juan Ahtziri 2012-06-25T20:11:28Z 2012-06-26T09:18:35Z <p>@Mariano: In my situation the function is not a Morse function and the curves of level are connected by paths, I think that this hypothesis about U are enough, but I can't proof this affirmation.</p> <p>@Anton: I don't know if the Whitehead manifold has a function with this properties.</p> <p>I want to proof that the compact sets $K_m = U-f^{-1}([0,m])$ (assuming that $0$ is the minimum value of $f$) are simply-connected, this will implies that $U$ is simply-connected to infinite.</p> http://mathoverflow.net/questions/100612/contractible-open-sets/100649#100649 Answer by Igor Rivin for contractible open sets Igor Rivin 2012-06-26T00:53:10Z 2012-06-26T00:53:10Z <p>See <a href="http://mathoverflow.net/questions/53841/hidden-convexity" rel="nofollow">http://mathoverflow.net/questions/53841/hidden-convexity</a> (almost, but not quite a duplicate)</p>