Open Boxes in ℝⁿ

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$.

(i) Open box. For $a,b\in\mathbb{R}^{n}$, the open box $R(a,b)$ is the subset

$$R(a,b)=\{x\in\mathbb{R}^{n}:a_{i}<x_{i}<b_{i}\text{ for every }i\in[n]\}$$

of $\mathbb{R}^{n}$ (a set, being a subclass of the set $\mathbb{R}^{n}$, by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)).

(ii) The open boxes of $\mathbb{R}^{n}$. The set of open boxes of $\mathbb{R}^{n}$ is

$$\mathcal{R}_{n}=\{R(a,b):a,b\in\mathbb{R}^{n}\}$$

(a set, being a class of subsets of $\mathbb{R}^{n}$, by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (iii) and Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)).