It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric spaces are topologically complete. The question is: there is any metric space that is locally compact and not topologically complete?

Another motivation for this question is: Hausdorff locally compact spaces are Baire spaces, as well topologicaly complete space. But there are metric spaces that are Baire, but not topologicaly complete. So, the natural question is that above.

Obs: A metric space $(X,d)$ is topologically complete if there is a metric $d′$ on $X$ such that $Id:(X,d) \mapsto (X,d′)$ is homomorphism and $(X,d′)$ is complete.

It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric spaces are topologically complete. The question is: there is any metric space that is locally compact and not topologically complete?

Another motivation for this question is: Hausdorff locally compact spaces are Baire spaces, as well topologicaly complete space. But there are metric spaces that are Baire, but not topologicaly complete. So, the natural question is that above.

Obs: A metric space $(X,d)$ is topologically complete if there is a metric $d′$ on $X$ such that $Id:(X,d) \rightarrow (X,d′)$ is a homeomorphism and $(X,d′)$ is complete.

It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric spaces are topologically complete. The question is: there is any metric space that is locally compact and not topologically complete?

Another motivation for this question is: Hausdorff locally compact spaces are Baire spaces, as well topologicaly complete space. But there are metric spaces that are Baire, but not topologicaly complete. So, the natural question is that above.

It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric spaces are topologically complete. The question is: there is any metric space that is locally compact and not topologically complete?

Another motivation for this question is: Hausdorff locally compact spaces are Baire spaces, as well topologicaly complete space. But there are metric spaces that are Baire, but not topologicaly complete. So, the natural question is that above.

Obs: A metric space $(X,d)$ is topologically complete if there is a metric $d′$ on $X$ such that $Id:(X,d) \mapsto (X,d′)$ is homomorphism and $(X,d′)$ is complete.