# Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property:

Every compatible metric (i.e one which induces the same topology $\tau$) is complete.

Compact metrizable spaces satisfy this.

Are there any non-compact examples?

Note:

It turns out that if you impose additional structure the answer is yes. In particular, if $M$ is a smooth manifold, and every Riemannian metric on $M$ is complete, then $M$ must be compact.

Reference: Nomizu, Katsumi, and Hideki Ozeki. "The existence of complete Riemannian metrics." Proceedings of the American Mathematical Society 12.6 (1961): 889-891.

http://www.jstor.org/discover/10.2307/2034383?uid=2&uid=4&sid=21105114015163is