Decompositions of Euclidean spaces by nontrivial continua. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:03:52Z http://mathoverflow.net/feeds/question/42790 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42790/decompositions-of-euclidean-spaces-by-nontrivial-continua Decompositions of Euclidean spaces by nontrivial continua. rpotrie 2010-10-19T14:18:22Z 2010-11-27T15:05:06Z <p>First, a brief introduction of the context. A <em>usc</em> decomposition of a metric space $X$ is a collection $G$ of compact sets which vary semicontinuously. I will be interested in <em>cellular</em> decompositions of $\mathbb{R}^n$ (that is, the elements of $G$ are decreasing intersections of closed balls). </p> <p>In 1967, <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.bams/1183529403" rel="nofollow">Jones proved</a> that it is not possible to fill $\mathbb{R}^n$ with disjoint arcs. This came after <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.dmj/1077489337" rel="nofollow">Roberts had proven</a> that the plane can be decomposed by cellular sets none of which is a point (or equivalently, there is a lower bound on the diameter of the sets). </p> <p>After some time searching the web, etc, I've found out that there has been a lot of progress towards the understanding of cellular decompositions of manifolds (see <a href="http://books.google.com/books?id=qyPRlNDbVksC&amp;pg=PA187&amp;lpg=PA187&amp;dq=daverman+decompositions+of+manifolds&amp;source=bl&amp;ots=QfiROu4AP4&amp;sig=ptShJY6SgSrpozg9I2pn7QbSD10&amp;hl=es&amp;ei=sKW9TIDfLJSI4gaD3unQAQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBQQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">Daverman's book</a>) but some kind of questions have been somewhat not treated. </p> <p>Is there some progress towards the understanding of cellular decompositions of $\mathbb{R}^n$ with all of the elements of diameter bounded from below? In particular, I am interested in the following questions:</p> <p>Q1) What is the possible topology of the sets constituting such a decomposition? Mainly, which kind of possible topologies are compatible, etc. </p> <p>In general (at least in dimension 2) such a decomposition is equivalent to the existence of a surjective continuous map $f:\mathbb{R}^n \to \mathbb{R}^n$ such that the preimage of every point is an element of $G$ (in particular, compact connected and cellular). </p> <blockquote> <p>So, assuming $f:\mathbb{R}^n \to \mathbb{R}^n$ is a surjective continuous map such that the preimage of each point is cellular and of diameter bigger than $0$</p> </blockquote> <p>Q2) What kind of preimages may arcs have under $f$? Or disks? What is the topology of this sets? </p> <p>And the question I am most interested in is:</p> <p>Q3) Consider an open set $U\in \mathbb{R}^n$ such that $f(U)$ is open: Does it hold that $U$ contains a fiber of $f$ (that is, a set of the form $f^{-1}(x)$)?</p> <p>Answers in dimension $2$ are already very interesting to me. Other related references will be appreciated. </p>