Open sets decompose countably into disjoint intervals in one dimension and almost disjoint closed cubes in higher dimensions.Suggest a slogan
Every open set \(U\subseteq\RR\) is a union of countably many pairwise disjoint open intervals, possibly unbounded, and these intervals are uniquely determined by \(U\): they are the connected components of \(U\).
For \(n\geq1\), every open set \(U\subseteq\RR^n\) is a union of countably many pairwise almost disjoint closed cubes [@SS05].
The decomposition in (b) is not unique: replacing one cube of such a family by the \(2^n\) closed cubes of half its side length obtained by bisecting each of its edges gives another countable family of pairwise almost disjoint closed cubes with union \(U\).