Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties 
[Wikipedia - Topological property][1] 

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.

  [1]: http://en.wikipedia.org/wiki/Topological_property