The sigma-algebra generated by the half-open boxes of ${\mathbb{R}}^n$ is the Borel sigma-algebra

Given: A natural number $n\ge 1$, the topology $\mathcal{T}$ of $({\mathbb{R}}^n,d_2)$, the family ${\mathcal{H}}_n$ of half-open boxes and the family ${\mathcal{E}}_n$ of elementary sets.

$B(a,b):=\{x\in {\mathbb{R}}^n:a_i<x_i\le b_i\text{ for every }i<n\}$, with parameters in $\bar{\mathbb{R}}$ (Half-open boxes in ${\mathbb{R}}^n$ and their volume).

A subset $E\subseteq {\mathbb{R}}^n$ is an elementary set when there are a natural number $m$ and a list $B_0,\dots ,B_{m-1}$ of half-open boxes with $E={\bigcup }_{j<m}{B}_j$; at $m=1$ every half-open box is elementary (Elementary sets: the finite unions of half-open boxes in ${\mathbb{R}}^n$).

Every open $U\subseteq {\mathbb{R}}^n$ is the union of an at most countable family of pairwise disjoint dyadic cubes (Every open subset of ${\mathbb{R}}^n$ is the union of a countable pairwise disjoint family of dyadic cubes), and a dyadic cube is the half-open box $Q_{k,m}=\{x:m_i{2}^{-k}<x_i\le (m_i+1)2^{-k}\text{ for every }i<n\}$ (Dyadic cubes of generation $k$ in ${\mathbb{R}}^n$).

Each of the following families generates $\mathcal{B}({\mathbb{R}}^n)$: all open sets; and all rational half-open boxes ${\prod }_{i<n}(a_i,b_i]$ with rational endpoints $a_i<b_i$ (For n at least one, open sets, closed sets, compact sets, open balls, boxes, rational open boxes, and rational half-open boxes generate the Borel sigma-algebra on R^n).

The Borel sigma-algebra of $X$ is the sigma-algebra generated by its open sets, $\mathcal{B}(X):={\sigma }_X(\mathcal{T})$ (The Borel sigma-algebra of a topological space), and ${\sigma }_X(\mathcal{E})$ is the unique smallest sigma-algebra on $X$ containing $\mathcal{E}$ (The sigma-algebra generated by a family of sets, Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal).

If $\mathcal{E}\subseteq {\sigma }_X(\mathcal{F})$ and $\mathcal{F}\subseteq {\sigma }_X(\mathcal{E})$, then ${\sigma }_X(\mathcal{E})={\sigma }_X(\mathcal{F})$ (Two families generate the same sigma-algebra when each lies in the sigma-algebra generated by the other).

A sigma-algebra is closed under countable unions and under countable intersections (Sigma-algebras, Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits).

The product topology on ${\mathbb{R}}^n$ is the metric topology of $d_2$ (A subset of ${\mathbb{R}}^n$ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology, claim 1; The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

A subset $U\subseteq X$ is open in $(X,d)$ if for every $x\in U$ there is a real $r>0$ with $B(x,r)\subseteq U$, and a finite intersection of open sets is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, claim 3).

For every $x\in {\mathbb{R}}^n$, $\parallel x{\parallel }_{\mathrm{\infty }}\le \parallel x{\parallel }_2$, and $\parallel x-y{\parallel }_2=d_2(x,y)$, $\parallel x-y{\parallel }_{\mathrm{\infty }}=d_{\mathrm{\infty }}(x,y)$, where $d_{\mathrm{\infty }}(x,y)=\max \{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}$ (The finite and reverse triangle inequalities for a norm; and for $n\ge 1$ every norm $N$ on ${\mathbb{R}}^n$ satisfies $N(x)\le C\parallel x{\parallel }_1$ and is Lipschitz, hence continuous, for $d_2$, claim 3; Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page, claim 3; The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$; ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it).

For every real $\epsilon >0$ there is a natural number $k\ge 1$ with $1\mathrm{/}k<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

For $a,b\in \bar{\mathbb{R}}$, $a+b:=+\mathrm{\infty }$ when $a=+\mathrm{\infty }$ and $b\mathrm{\ne }-\mathrm{\infty }$, and $a+b:=-\mathrm{\infty }$ when $a=-\mathrm{\infty }$ and $b\mathrm{\ne }+\mathrm{\infty }$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined).

An at most countable family may always be presented as a sequence (Finite, countably infinite, countable, uncountable).