gn.general topology - What are the topological properties of the metric space retained (inherited) for its completion

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.