Every Polish space is countable or of cardinality $\mathfrak{c}$.

Consequently, every countable Polish space has a homeomorphic copy with underlying set the natural numbers, and every uncountable one has a copy with $\mathbb{R}$ as its underlying set. In the latter case the bijection can be made a Borel isomorphism (see Theorem 15.6 in Kechris' Classical Descriptive Set Theory.

Every Polish space is countable or of cardinality $\mathfrak{c}$.

Consequently, every countable Polish space has a homeomorphic copy with underlying set the natural numbers, and every uncountable one has a copy with $\mathbb{R}$ as its underlying set. In the latter case the bijection can be made a Borel isomorphism (see Theorem 15.6 in Kechris' Classical Descriptive Set Theory).