I have the following exercise that I wish to solve:
Let $U\subseteq\mathbb{R}^{n}$ be an open set, $T\subseteq\mathbb{R}^{n}$ will be called a box if its of the form $[a_{1},b_{1}]\times\ldots[a_{n},b_{n}]$ where $a_{i},b_{i}\in\mathbb{R}$.
Show that there exist a sequence $\{T_{k}\}_{k=1}^{\infty}$if boxes with disjoint interior (they intersect, at most, at their sides) s.t $U=\cup_{k=1}^{\infty}T_{k}$
Explain why $U$ can not be a partition of a finite number of boxes
Show that the open unit ball in $\mathbb{R}^{2}$ can not be written as a disjoint countable union if sets of the form $(a_{1},b_{1})\times(a_{2},b_{2})$.
My work:
For the first part of the question I tried to partition $\mathbb{R}^{n}$ into boxes of sizes $r\times r$ and take those boxes that are contained in $U$. I hoped that for a small enough $r$ this set will be $U$, since $U$ is open for any $u\in U$ there is some $r'$ s.t $B(u,r')\subseteq U$ so I wanted $r
For $2$ I think that such a partitioning means $U$ is also closed hence each $[a_{i},b_{i}]=(-\infty,\infty)$ but its clear that you can't partition $(-\infty,\infty)$ in the described way.
I have no clue on how to start $3$.
I would appriciate any help on doing this exercise!