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. 



Scattered separators in Erdős space
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 $|S|\ge|U\cap X\_1|=\mathfrak c$.
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