From this question I realize that there exists an open set in $\mathbb{R}^2$ that is not a disjoint union of open rectangles. The example given is the set of points lying below the line $y=-x$.
However, I can't quite see how one would prove that that particular set is not a disjoint union of open rectangles. What contradiction can one derive if that set were to be a disjoint union of open rectangles?
Best Answer
For one thing, a disjoint union of a family of open sets can only be connected if at most one member of the family is non-empty. The half-plane $\{y < -x\}$ is connected and not an open rectangle, hence it cannot be the disjoint union of open rectangles.
Another way to see it is to note that in a disjoint union of open rectangles, no boundary point of any rectangle can be covered. For any open set containing the boundary point must intersect the rectangle of which it is a boundary point.