# Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$

I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite difficult

I am aware that a lot of posts already exists in this direction, for example this one: Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] But it is always about intervals on $\mathbb{R}$ and the proofs are all fairly tough.

Does anyone know if there exists a reference of this proof that is sufficiently easy for beginners in analysis?

• I suppose it is, though natural is of course a subjective term. It's important to note that only in $\mathbb{R}$ can the intervals be made disjoint; in higher dimensions "almost disjoint" is the best you can do. Apr 26 '16 at 2:29
• In higher dimensions, if you use intervals of the form $(a_1,b_1]\times\cdots(a_n,b_n]$ then they can be made disjoint. This is useful when applying countable additivity of measures.