Exercise 7, § 6

a) Show that every zero-dimensional Polish space is homeomorphic to a closed subspace of the product $J = \mathbf{N}^\mathbf{N}$ (consider a strict sifting of X by sets which are both open and closed).

b) Let X be a zero-dimensional Polish space and let Y be a dense subspace of X which has no interior point and which is a countable intersection of open sets in X. Show that Y is homeomorphic to J. (Note that, in a metrizable zero-dimensional space, an open but not closed set is the union of an infinite countable sequence of pairwise disjoint sets which are both open and closed; use Theorem 1 of no. 1.) Deduce that every dense subspace of R, which has no interior point and is a countable intersection of open sets, is homeomorphic to J (note that such a set is contained in the complement of a countable dense subset D of R, and that $\mathbf{C}D$ is zero-dimensional).

c) If X is any uncountable zero-dimensional Polish space, there exists a partition of X into a countable set and a subspace homeomorphic to J [use b) and Exercise 11 of § 5].