Open Boxes: Emptiness, Intersections and Preimages under Concatenation

In the setting of Euclidean Space and Real Matrices: Standing Notation, for $n\in\mathbb{N}$ and $a,b\in\mathbb{R}^{n}$ let $R(a,b)$ be the open box with corners $a$ and $b$. For $s,t\in\mathbb{R}$, $\max\{s,t\}$ and $\min\{s,t\}$ are as in The Maximum and Minimum of Two Elements of a Total Order (i); for a function $u:S\to T$ and $V\subseteq T$, $u^{-1}(V)$ is the preimage of $V$, as in Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set.

(i) Empty boxes. Let $n\in\mathbb{N}$ and $a,b\in\mathbb{R}^{n}$. If $b_{i}\le a_{i}$ for some $i\in[n]$, then $R(a,b)=\emptyset$.

(ii) Intersections. Let $n\in\mathbb{N}$ and $a,b,a',b'\in\mathbb{R}^{n}$. There are $c,d\in\mathbb{R}^{n}$ with $c_{i}=\max\{a_{i},a'_{i}\}$ and $d_{i}=\min\{b_{i},b'_{i}\}$ for every $i\in[n]$, and

$$R(a,b)\cap R(a',b')=R(c,d).$$

(iii) Preimages under concatenation. Let $m,n\in\mathbb{N}$, let $\iota:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{m+n}$ be the concatenation function of Euclidean Space and Real Matrices: Standing Notation (viii), let $a,b\in\mathbb{R}^{m+n}$, and let $a',b'\in\mathbb{R}^{m}$ and $a'',b''\in\mathbb{R}^{n}$ with $\iota(a',a'')=a$ and $\iota(b',b'')=b$. Then

$$\iota^{-1}\bigl(R(a,b)\bigr)=R(a',b')\times R(a'',b'').$$