Polish space - Knowlpedia

A Polish space is a separable topological space $X$ for which there exists a metric $d$ whose induced topology is the given topology and such that $(X,d)$ is a complete metric space. Completeness is required for at least one compatible metric; it need not hold for every metric inducing the topology.

Examples

$\mathbb R^n$ , with its usual topology, is Polish for every integer $n\ge0$ : the Euclidean metric is complete and $\mathbb Q^n$ is a countable dense subset.
Every open interval $(a,b)\subset\mathbb R$ , where $a<b$ are real numbers, is Polish. Its usual metric is not complete, but a homeomorphism $h:(a,b)\to\mathbb R$ supplies the compatible complete metric $d_h(x,y)=|h(x)-h(y)|$ .
$\mathbb Q$ with its usual topology is not Polish. It is separable, but has no isolated points and is a countable union of nowhere-dense singleton sets, hence is meagre in itself. A nonempty completely metrizable space is Baire, so no compatible metric on this topology can be complete.

Remarks

Polishness is a property of the topology, not of a selected metric. It is a standard state-space hypothesis in probability and descriptive set theory.