Exercise 9, § 6

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

b) In a scattered Polish space $X$, let $Y$ be an everywhere dense subspace, without interior point and countable intersection of open sets in $X$. Prove that $Y$ is homeomorphic to $J$. (Remark that in a scattered metrizable space an open non-closed set is the union of an infinite countable sequence of sets which are both open and closed, pairwise disjoint; use th. 1 of IX, p. 58). Deduce that in $\mathbf{R}$ every everywhere dense subspace, without interior point and countable intersection of open sets, is homeomorphic to $J$ (remark that such a set is contained in the complement of a countable everywhere dense subset $D$ of $\mathbf{R}$, and that $\complement D$ is scattered).

c) For every uncountable scattered Polish space $X$, there exists a partition of $X$ into a countable set and a subspace homeomorphic to $J$ (Use $b$) and I, p. 108, Exercise 17).