Can someone please tell me where I can find a citeable reference for the following result:


Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which is complete under $d$.


If $(X,d)$ is locally complete and separable then there exists a metric $d'$ on $X$ such that $(X,d) \to (X,d')$ is a homeomorphism and $(X, d')$ is complete.


This result follows immediately from Alexandrov's theorem that a $G\_\delta$ subset of a Polish space is Polish, but I'd rather find a statement in the literature of the above more straightforward result if there is one.



Locally complete space is topologically equivalent to a complete space
For a reference: [this paper](http://www.chimiefs.ulg.ac.be/SRSL/newSRSL/modules/FCKeditor/upload/File/73_1/Bella%20TSIRULNIKOV%20p%209-19.pdf) has a reference [30] that has a proof. The author cites your result and refers to it. I don't have access to these papers, so I cannot verify exactly.


