What are the topological properties of the metric space retained (inherited) for its completion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:28:49Z http://mathoverflow.net/feeds/question/37289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37289/what-are-the-topological-properties-of-the-metric-space-retained-inherited-for What are the topological properties of the metric space retained (inherited) for its completion Ivan Gundyrev 2010-08-31T18:17:04Z 2010-09-01T05:41:16Z <p>Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion. There is a list of topological properties <a href="http://en.wikipedia.org/wiki/Topological_property" rel="nofollow">Wikipedia - Topological property</a> </p> <p>Does anybody know list which of them are retained (inherited) for completion? For example</p> <ol> <li><p>if $(X,d)$ is locally compact space then $\bar{X}$ may be non-locally compact space.(Consider the induced path metric space on the following subset of the Euclidean plane: $(0,1] \times { 0 }\cup (0,1]\times {1}\cup \bigcup_{n=1}^{\infty}{1/n}\times [0,1]$.) </p></li> <li><p>if $(X,d)$ is separable space then $\bar{X}$ is separable space.</p></li> <li>if $(X,d)$ is connected space then $\bar{X}$ is connected space.</li> <li>if $(X,d)$ is path-connected space then $\bar{X}$ may be non path-connected space. (consider the graph of $\sin(1/x)$ in the plane for positive $x$. )</li> </ol> <p>I am interested in this problem in general, especially for the spaces with intrinsic metric.</p> http://mathoverflow.net/questions/37289/what-are-the-topological-properties-of-the-metric-space-retained-inherited-for/37348#37348 Answer by Elizabeth S. Q. Goodman for What are the topological properties of the metric space retained (inherited) for its completion Elizabeth S. Q. Goodman 2010-09-01T05:12:46Z 2010-09-01T05:41:16Z <p>I was going to suggest that all the connectivity properties were either preserved or sometimes acquired by completion: e.g. a totally disconnected $X$ may become a path-connected $\bar X$, and the same is true for the other locally-defined connectivity properties I considered on that list. </p> <p>But this is not true in the case of simple connectivity, or n-connectivity, because these properties depend on each point. As far as I can tell you can change them any way you like. You could put a metric on a CW-complex, but for $\bar X$ any CW complex of countably many cells, you can remove a point to change the homotopy type of $X$ as compared to $\bar X$, or just as above, let $X$ be a discrete dense set.</p> <p>Or make $X$ two horizontal line segments one over the other, connected by line segments depicting an ordered bijection between dense subsets, or higher-dimensional analogues, so that $\bar X$ is a cube.</p> <p>Or let $X$ be the cone of any topological space with an appropriate metric, but with the point at the tip removed, so whatever the homotopy type of $X$, $\bar X$ is contractible. I think you could even selectively remove points from a CW complex to redesign homotopy groups in more interesting ways.</p>