What does the property that path-connectedness implies arc-connectedness imply? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:32:35Z http://mathoverflow.net/feeds/question/1406 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1406/what-does-the-property-that-path-connectedness-implies-arc-connectedness-imply What does the property that path-connectedness implies arc-connectedness imply? skupers 2009-10-20T09:51:26Z 2012-08-03T14:25:52Z <p>A space X is path-connected if any two points are the endpoints of a path, that is, the image of a map [0,1] \to X. A space is arc-connected if any two points are the endpoints of a path, that, the image of a map [0,1] \to X which is a homeomorphism on its image. If X is Hausdorff, then path-connected implies arc-connected.</p> <p>I was wondering about the converse: What properties must X have if path-connected implies arc-connected? In particular, what are equivalent properties?</p> http://mathoverflow.net/questions/1406/what-does-the-property-that-path-connectedness-implies-arc-connectedness-imply/1423#1423 Answer by Harald Hanche-Olsen for What does the property that path-connectedness implies arc-connectedness imply? Harald Hanche-Olsen 2009-10-20T13:22:08Z 2009-10-20T13:22:08Z <p>I don't have an answer, but here is an example to show it's not a local property that decides it. Consider the real line with two inseparable zeros, 0 and 0'. Clearly there is a path from 0 to 0' but not an arc. On the other hand, if you adjoin a point at infinity, making a circle with a double point on it, you can make such an arc going through infinity, and so the space is arc-connected.</p> http://mathoverflow.net/questions/1406/what-does-the-property-that-path-connectedness-implies-arc-connectedness-imply/25483#25483 Answer by KP Hart for What does the property that path-connectedness implies arc-connectedness imply? KP Hart 2010-05-21T12:59:40Z 2010-05-21T12:59:40Z <p>It suffices that $X$ be Hausdorff: the path is then a compact metric image of [0,1] and as such arc-wise connected (do Problem 6.3.11 of Engelking's General Topology). </p>