Wikipedia gives as an example for Polish spaces the open interval $(0, 1)$. Can somebody explain to me how $(0,1)$ can be Polish?
$(0, 1)$ has to be metrizable so that it is complete, which means that the sequence $\left(\frac{1}{n}\right)_{n\in\mathbb{N}}$ must not converge to zero anymore using this metric.
You think about $(0,1)$ with the standard metric. But think about it with a different metric.
If you already know that $(0,1)$ is homeomorphic with $\Bbb R$ this gives you a natural metric. Just pull back the metric using some homeomorphism.
More generally, a subset of Polish space is Polish if and only if it is a $G_\delta$ set there. So it's not only that $(0,1)$ is a Polish space, $\Bbb{R\setminus Q}$ is also a Polish space.
