Polish space

Remarks

The real line $\R$ is Polish: the rationals are dense, and the usual metric is complete. The open interval $(0, 1)$ is Polish, because the metric $d(x, y) = \abs{\tan(\pi x - \pi/2) - \tan(\pi y - \pi/2)}$ induces its topology and is complete. The rationals $\Q$ are a non-example: they are separable and metrizable, but no complete metric induces their topology. Indeed, by the Baire category theorem a nonempty complete metric space is not a countable union of nowhere dense sets, while $\Q$ is the countable union of its points. An uncountable discrete space is another non-example: it is completely metrizable but not separable. A subset of a Polish space is Polish in the subspace topology if and only if it is a $G_\delta$ set.