I know that completeness itself is not a topological property because a complete and a not complete metric space can be homeomorphic, e.g. $\Bbb R$ and $(0,1)$.
However, both $\Bbb R$ and $(0,1)$ are locally complete (each point has a neighborhood that is complete under the induced metric). As all examples I know of are of this form, the naturally occuring next question is
Question: Is being locally complete a topological property?
Or the other way around: are there metric spaces which are homeomorphic, but one is locally complete and the other one is not?