Is locally completeness a topological property? 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?

The irrational numbers are not locally complete, but they are homeomorphic to the Baire space $\mathbb N^{\mathbb N}$, which can be given a metric turning it into a complete metric space.  
(For example, endow $\mathbb N$ with the discrete metric and set $d(s, t) = \sum_{i=0}^\infty\frac{1}{2^i}d(s_i,t_i)$).  
