A density condition for metric spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:11:22Z http://mathoverflow.net/feeds/question/66268 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66268/a-density-condition-for-metric-spaces A density condition for metric spaces Valerio Capraro 2011-05-28T10:06:29Z 2011-09-19T16:22:12Z <p>I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?</p> <blockquote> <p><strong>Property:</strong> $(X,d)$ metric space. For any open ball $B\subseteq X$ and for any distinct $x,y\in B$, there exist two disjoint open balls $B_1\ni x$ and $B_2\ni y$ and two open continuous and injective functions $f_i:B\rightarrow X$ such that $f_i(B)\subseteq B_i$.</p> </blockquote> <p>Well.. it's similar to contractibility, but seems to be <em>weaker</em> - it's some density condition.. </p> http://mathoverflow.net/questions/66268/a-density-condition-for-metric-spaces/66365#66365 Answer by Sergey Melikhov for A density condition for metric spaces Sergey Melikhov 2011-05-29T14:45:09Z 2011-05-30T11:28:51Z <p>An open continuous injective map can also be described as a homeomorphism with an open subset. The stated property is obviously equivalent to: every open set in every open ball $B$ in $X$ contains an open set that is homeomorphic with $B$. If the metric is bounded, $B$ can be replaced by $X$ without loss of generality (and so the property becomes purely topological).</p> <p>This property is a kind of <em>local self-similarity</em> (or just self-similarity in the bounded case). </p> <p>The property does not hold for contractible nor for locally contractible spaces, for instance it fails already for the closed unit interval. It fails even "locally" for the triod, in the sense that the vertex of the triod has no neighborhood satisfying the property.</p>