Scattered separators in Erdős space

Question

Let $X$ be the set of all points in $\ell^2$ with all rational coordinates. $X$ is known to be totally disconnected, but $X$ is not zero-dimensional. For instance, the empty set does not separate the point $\langle 0,0,0,...\rangle\in X$ from the closed set $\{x\in X:\|x\|\geq 1\}$ because $\{\|x\|:x\in A\}$ is unbounded for every clopen set $A\subseteq X$.

The set $S:=\{x\in X:\|x\|=1/2\}$ separates $\langle 0,0,0,...\rangle$ and $\{x\in X:\|x\|\geq 1\}$. That is, $X\setminus S$ is the union of two disjoint open sets, one containing $\langle 0,0,0,...\rangle$, and the other containing $\{x\in X:\|x\|\geq 1\}$. Note that $S$ has no isolated points; $\overline {S\setminus \{s\}}=S$ for every $s\in S$.

My question is:

Does there exist a closed scattered separator between $\langle 0,0,0,...\rangle$ and $\{x\in X:\|x\|\geq 1\}$? A set is scattered if every non-empty subset has an isolated point.

Answer 1

Observe that scattered subsets in metrizable separable (more generallity hereditarily Lindelof) spaces are at most countable.

On the other hand, any closed separator $S$ between zero and the set $F=\{x\in X:\|x\|\ge 1\}$ has cardinality continuum and hence cannot be scattered.

To see that $|S|=\mathfrak c$, write $X\setminus S$ as the union $U\cup V$ of two disjoint open sets $U,V$ such that $0\in U$ and $F\subset V$. For $i\in\{0,1\}$ consider the closed subspace $X_i=\{(x_n)_{n\in\omega}\in X:\forall n\in\omega\; x_{2n+i}=0\}$ and observe that $X=X_0+X_1$ and $|U\cap X_1|=\mathfrak c$.

Let $(e_i)_{i\in\omega}$ be the standard orthonormal basis of the Hilbert space $\ell_2$. For every $u\in U\cap X_1$ we have $\|u\|<1$ and $u+e_0\in F\subset V$. Consequently, there exist rational numbers $x_0<x_0'$ such that $u+x_0e_0\in U$, $u+x_0'e_0\in V$, and $|x_0'-x_0|<1$. Since $u+x_0e_0+e_2\in A\subset V$, there exist $x_2<x_2'$ such that $u+x_0e_0+x_2e_2\in U$, $u+x_0e_0+x_2'e_2\in V$ and $|x_2'-x_2|<1/4$. Continuing by induction we shall construct two sequences of rational numbers $(x_{2i})_{i\in\omega}$ and $(x_{2i}')_{i\in\omega}$ such that for every $n\in\omega$ $$u+\sum_{i=0}^nx_{2i}e_{2i}\in U,$$ $$u+\sum_{i=0}^nx_{2i}e_{2i}+(x_{2n}'-x_{2n})e_{2n}\in V\mbox{; and }$$$$|x_{2n}'-x_{2n}|<1/2^{2n}.$$ Since $U\subset\{x\in \ell_2:\|x\|<1\}$, the series $\sum_{i=0}^\infty x_{2i}e_{2i}$ is convergent and the point $s(u)=u+\sum_{i=0}^\infty x_{2i}e_{2i}$ belongs to the separatior $S$.

Since the map $s:U\cap X_1\to S$, $s:u\mapsto s(u)$, is injective, the separator $S$ has cardinality $|S|\ge|U\cap X_1|=\mathfrak c$. $\;\;\;\;\;$