MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Post Made Community Wiki by Ivan Gundyrev
4 add example for locally compact case

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: $(0,1]\times {0} \cup (0,1] (0,1] \times { 1} \cup \bigcup_{n=1}^{\infty} 0 }\cup (0,1]\times {1/n} 1}\cup \times bigcup_{n=1}^{\infty}{1/n}\times [0,1]$.) 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)$\sin(1/x)$ in the plane for positive $x$. )

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

3 add example for locally compact case

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.

2 changed when X is path-connected
1