# Open set of Polish space is again a Polish space?

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?

• $(0,1)$ is homeomorphic to $\mathbb R$, which is completely metrizable. – user2345215 Nov 15 '14 at 10:29
• A space homeomorphic to a Polish space is Polish. It's a topological definition, not a metric one. – Henno Brandsma Nov 15 '14 at 10:49

In fact, every $G_\delta$ subset of a Polish space is Polish; and if $X$ is a Polish space, and $Y\subseteq X$ is a Polish space (with the induced topology from $X$), then we can prove that $Y$ is a $G_\delta$ subset of $X$.
• I believe separability has nothing to do with it; i.e., a subset of a completely metrizable space is completely metrizable if and only if it's a $G_\delta$. – bof Nov 15 '14 at 10:39