A Polish space is a separable completely metrizable topological space. On the wikipidia article and in the book measure theory from Bauer (§26, Example 4) is stated that any open set of a polish space is again a polish space.
In the book from Bauer one can find a proof that shows that any open set of a Polish space is homeomorphic to another Polish space. But how is it possible that a open set is again completely metrizable? For instance, the set $[0,1]$ is Polish and $(0,1)$ is an open subset which is not completely metrizable, right?