1
$\begingroup$

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.

$\endgroup$
10
  • $\begingroup$ You mean, which properties are retained when passing from a space to its completion? $\endgroup$ Aug 31, 2010 at 18:52
  • 2
    $\begingroup$ Could you please give some examples of properties you're interested in and for which you don't know the answer? $\endgroup$ Aug 31, 2010 at 19:18
  • 4
    $\begingroup$ Example 4 is not true: consider the graph of $\sin(1/x)$ in the plane for positive $x$. $\endgroup$ Aug 31, 2010 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. $\endgroup$ Aug 31, 2010 at 21:52
  • $\begingroup$ @Joel, Pete Point taken. My mistake. $\endgroup$ Aug 31, 2010 at 22:07

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.