The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates

In the settings of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation and Euclidean Space and Real Matrices: Standing Notation, let $n,p,q\in\mathbb{N}$. $\mathbb{R}^{n}$ carries the Euclidean distance $d_{E}$ and $\mathcal{M}_{p\times q}(\mathbb{R})=\mathbb{R}^{p\times q}$ the distance $d_{p\times q}$, as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (vi); $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R}^{p\times q})$ are the Borel $\sigma$-algebras of Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii) and Borel Sigma-Algebra of a Metric Space (iv). For $i\in[n]$, $\pi_{i}:\mathbb{R}^{n}\to\mathbb{R}$ is the $i$-th component map of the identity $\mathrm{id}_{\mathbb{R}^{n}}$, so $\pi_{i}(x)=x_{i}$; for $i\in[p]$ and $j\in[q]$, $E_{ij}:\mathbb{R}^{p\times q}\to\mathbb{R}$ is the entry map of $\mathrm{id}_{\mathbb{R}^{p\times q}}$ in row $i$ and column $j$, so $E_{ij}(A)=A_{ij}$. $\mathbb{Q}^{n}$ is the set of $n$-tuples in $\mathbb{Q}$, whose components are read as real numbers as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (v); for $a,b\in\mathbb{Q}^{n}$, the rational open box

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

is a subset of $\mathbb{R}^{n}$. Likewise $(\mathbb{Q}^{q})^{p}$ is the set of $p$-tuples in $\mathbb{Q}^{q}$; for $a\in(\mathbb{Q}^{q})^{p}$, $i\in[p]$ and $j\in[q]$, $a_{ij}$ is the $j$-th component of the $i$-th component $a_{i}\in\mathbb{Q}^{q}$ of $a$, read as a real number; and for $a,b\in(\mathbb{Q}^{q})^{p}$, the rational open entry box

$$R'(a,b)=\{A\in\mathbb{R}^{p\times q}:a_{ij}<A_{ij}<b_{ij}\text{ for all }i\in[p],\ j\in[q]\}$$

is a subset of $\mathbb{R}^{p\times q}$.

(i) For every subset $U$ of $\mathbb{R}^{n}$ that is open in $(\mathbb{R}^{n},d_{E})$ there is a sequence $(W_{k})_{k\in\mathbb{N}}$ of subsets of $\mathbb{R}^{n}$ with $U=\bigcup_{k\in\mathbb{N}}W_{k}$ such that, for every $k\in\mathbb{N}$, $W_{k}\subseteq U$ and $W_{k}=R(a,b)$ for some $a,b\in\mathbb{Q}^{n}$.

(ii) For every subset $U$ of $\mathbb{R}^{p\times q}$ that is open in $(\mathbb{R}^{p\times q},d_{p\times q})$ there is a sequence $(W_{k})_{k\in\mathbb{N}}$ of subsets of $\mathbb{R}^{p\times q}$ with $U=\bigcup_{k\in\mathbb{N}}W_{k}$ such that, for every $k\in\mathbb{N}$, $W_{k}\subseteq U$ and $W_{k}=R'(a,b)$ for some $a,b\in(\mathbb{Q}^{q})^{p}$.

(iii) $\mathcal{B}(\mathbb{R}^{n})=\sigma(\{\pi_{i}^{-1}(B):i\in[n],\ B\in\mathcal{B}(\mathbb{R})\})$.

(iv) $\mathcal{B}(\mathbb{R}^{p\times q})=\sigma(\{E_{ij}^{-1}(B):i\in[p],\ j\in[q],\ B\in\mathcal{B}(\mathbb{R})\})$.