Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion. There is a list of topological properties Wikipedia - Topological property

Does anybody know list which of them are retained (inherited) for completion? For example

  1. 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] $.)

  2. if $(X,d)$ is separable space then $\bar{X}$ is separable space.

  3. if $(X,d)$ is connected space then $\bar{X}$ is connected space.
  4. 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 $x$. )

I am interested in this problem in general, especially for the spaces with intrinsic metric.

  • $\begingroup$ You mean, which properties are retained when passing from a space to its completion? $\endgroup$ – Stefan Geschke Aug 31 '10 at 18:52
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    $\begingroup$ Could you please give some examples of properties you're interested in and for which you don't know the answer? $\endgroup$ – Jonas Meyer Aug 31 '10 at 19:18
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    $\begingroup$ Example 4 is not true: consider the graph of $\sin(1/x)$ in the plane for positive $x$. $\endgroup$ – Joel David Hamkins Aug 31 '10 at 20:58
  • $\begingroup$ Some of these are trivial (or at least very elementary). For instance, 2 is true because any countable dense subset of $X$ is a countable dense subset of $\overline{X}$, since $X$ is dense in its completion. Similarly, anything between a connected set and its closure in a topological space is connected. In particular, if $X$ is connected then so is its completion. You probably should have thought a little bit harder about some of these. $\endgroup$ – Keenan Kidwell Aug 31 '10 at 21:52
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    $\begingroup$ community wiki? $\endgroup$ – David Corwin Sep 1 '10 at 4:18

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.

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.

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.

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.


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