Theorem: Every open subset $O$ of $R^{d}$, $d\geq1$ can be written as a countable union of almost disjoint closed cubes.
Let's take an example for $d=1$. We take the interval $(0,1)$. If I understand correctly, we say that the union of two closed intervals $[0,0.5]$ and $[0.5,1]$ represents our interval.
These two intervals are almost disjoint because there interior points are disjoint. Theire union is $[0,1]$.
But our interval $(0,1)$ and the union $[0,1]$ are different intervals. I know that the length is the same. But theorem say nothing about the length.
Where am I wrong?
To write $(0, 1)$ as a (countable) union of almost disjoint closed intervals, you need infinitely many closed intervals, since a finite union of closed intervals is a closed set. However, the almost disjoint intervals $[1/(n+1), 1/n]$, $[1 - 1/n, 1 - 1/(n+1)] $, for $n = 2, 3, \ldots$ do cover $(0, 1)$. Read the proof of the theorem to see how this works out in general for an arbitrary open subset of $\Bbb{R}^d$.