# General Topology – Scattered Separators in Erdös Space

Let $$X$$ be the set of all the points $$ell ^ 2$$ with all the rational coordinates. $$X$$ is known to be totally disconnected but $$X$$ is not zero-dimensional. For example, the empty game 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 unlimited for each clopen game $$A subseteq X$$.

L & # 39; together $$S: = {x in X: | x | = 1/2 }$$ separate $$langle 0,0,0, … rangle$$ and $${x in X: | x | geq 1 }$$. C & # 39; is, $$X setminus S$$ is the union of two disjoint open sets, one containing $$langle 0,0,0, … rangle$$and the other container $${x in X: | x | geq 1 }$$. Note that $$S$$ There are no isolated points; $$overline {S setminus {s }} = S$$ for each $$s in S$$.

My questions are:

(1) Is there a closed dispersed separator between $$langle 0,0,0, … rangle$$ and $${x in X: | x | geq 1 }$$? A set is scattered if each non-empty subset has an isolated point.

(2) Is there a point $$x in X$$ and a closed set $$A subseteq X setminus {x }$$ which can not be separated by a closed scattered ensemble?

(3) Repeat (2) for the full space $$Y$$ made up of all the points of $$ell ^ 2$$ who only have irrational coordinates.