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.