What are the topological properties of the metric space retained (inherited) for its completion 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


*

*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] $.) 

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

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

*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.
 A: 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.
