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

By Component Maps and Entry Maps (i) and Component Maps and Entry Maps (ii), $\pi_{i}(x)=x_{i}$ and $E_{ij}(A)=A_{ij}$.

The two cases. One argument serves the vectors and the matrices. In case V let $Y=\mathbb{R}^{n}$, $d=d_{E}$, $J=[n]$, $c_{j}=\pi_{j}$ for $j\in J$, $P=\mathbb{Q}^{n}$ and $m=n$, and for $a\in P$ and $j\in J$ let $a_{j}$ be the $j$-th component of $a$. In case M let $Y=\mathbb{R}^{p\times q}$, $d=d_{p\times q}$, $J=[p]\times[q]$, $c_{(i,j)}=E_{ij}$ for $(i,j)\in J$, $P=(\mathbb{Q}^{q})^{p}$ and $m=pq$, and for $a\in P$ and $(i,j)\in J$ let $a_{(i,j)}=a_{ij}$, as in the statement. In both cases, for $a,b\in P$ let

$$R_{Y}(a,b)=\{y\in Y:a_{j}<c_{j}(y)<b_{j}\text{ for every }j\in J\},$$

which is $R(a,b)$ in case V and $R'(a,b)$ in case M. As $n,p,q\in\mathbb{N}$, we have $0<n$, $0<p$ and $0<q$ in $\mathbb{R}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi), so $m$ is positive: $m=n$ in case V, and $m=pq$ with $0<pq$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (iv) in case M. The index set $J$ is countable and nonempty: $[n]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), since $k\le n$ for every $k\in[n]$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), likewise $[p]$ and $[q]$, so $[p]\times[q]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v), and finite sets are countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (iii); and $1\in[n]$ and $(1,1)\in[p]\times[q]$. Let $a,b\in P$. By Indexed Families of Sets and Their Union, Intersection and Product (iii), The Image and the Preimage of a Class under a Class (ii) and Intervals of the Real Line (ii), $J$ being nonempty, a set $y$ lies in $\bigcap_{j\in J}c_{j}^{-1}((a_{j},b_{j}))$ if and only if $y\in Y$ and $a_{j}<c_{j}(y)<b_{j}$ for every $j\in J$; so

$$R_{Y}(a,b)=\bigcap_{j\in J}c_{j}^{-1}((a_{j},b_{j})).$$

This expression gives the maps used below. The map $j\mapsto c_{j}$ is a map from $J$ to the set $\mathbb{R}^{Y}$ of maps $Y\to\mathbb{R}$: in case V it is the map $k\mapsto\pi_{k}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (i), applied to $\mathrm{id}_{\mathbb{R}^{n}}$, and in case M the map $(i,j)\mapsto E_{ij}$ of Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (ii), applied to $\mathrm{id}_{\mathbb{R}^{p\times q}}$. With $(a,b)\in P\times P$ and $j\in J$ varying, $a$ and $b$ are the values at $(a,b)$ of the two projection maps of $P\times P$ (Projection, Evaluation and Currying Maps (i)), and $a_{j}$ and $b_{j}$ are the values of the tuples $a$ and $b$ at $j$ under the evaluation map (Projection, Evaluation and Currying Maps (ii)), real numbers; in case M, for $j=(i,l)$, the indices $i$ and $l$ are the values of the projection maps of $[p]\times[q]$ at $j$, and $a_{il}=(a(i))(l)$ is obtained by evaluating twice, in the same way, and likewise $b_{il}$. So $(a_{j},b_{j})$ is a term of type $\mathcal{P}(\mathbb{R})$ (Intervals of the Real Line as Defined Set Symbols, Typed Terms and Values of Maps (ii)), and the intersected set $c_{j}^{-1}((a_{j},b_{j}))$ is a term of type $\mathcal{P}(Y)$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (i)), both with $(a,b)$ and $j$ varying. Hence $j\mapsto c_{j}^{-1}((a_{j},b_{j}))$ is a map from $J$ to the power set $\mathcal{P}(Y)$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (vi)), and $(a,b)\mapsto R_{Y}(a,b)$ is a map from $P\times P$ to $\mathcal{P}(Y)$ (Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (vi), Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (ii)).

Step 1 (the set $P$ is countable). For a set $S$ and $k\in\mathbb{N}$, the elements of $S^{k}$ are the maps $[k]\to S$, by Tuples in a Set: the Set of n-Tuples and Their Components (ii), so $S^{k}$ is countable whenever $S$ is, by Finite Powers of a Countable Set Are Countable (i). As $\mathbb{Q}$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix), $\mathbb{Q}^{n}$ and $\mathbb{Q}^{q}$ are countable, and then, with $S=\mathbb{Q}^{q}$, so is $(\mathbb{Q}^{q})^{p}$. So $P$ is countable in both cases.

Step 2 (two estimates). For all $y,z\in Y$: (E1) $|c_{j}(y)-c_{j}(z)|\le d(y,z)$ for every $j\in J$; (E2) if $t\in\mathbb{R}$ and $|c_{j}(y)-c_{j}(z)|\le t$ for every $j\in J$, then $d(y,z)\le mt$. In case V, $(y-z)_{j}=y_{j}-z_{j}$ for $j\in[n]$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), and $d_{E}(y,z)=\lVert y-z\rVert$ by The Euclidean Distance on $\mathbb{R}^n$ (i); so (E1) and (E2) are Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv) applied to $y-z$. In case M, $(y-z)_{kl}=y_{kl}-z_{kl}$ for $k\in[p]$ and $l\in[q]$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv), and $d_{p\times q}(y,z)=\lVert y-z\rVert$ by The Distance between Real Matrices (i). Then (E1) is The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (ii) applied to $y-z$; and under the hypothesis of (E2), by The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices (iv), Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (v) and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (iv), applied first to the inner and then to the outer sum, the differences $y_{kl}-z_{kl}$ being real numbers (Differences, Quotients and Absolute Values of Real Numbers, and Their Order in the Extended Real Line, Read in the Type Real (i)) and $p\,(qt)=(pq)\,t$ (The Real Numbers as a Type (iv)),

$$\lVert y-z\rVert\le\sum_{k=1}^{p}\sum_{l=1}^{q}|(y-z)_{kl}|\le\sum_{k=1}^{p}qt=p\,(qt)=mt .$$

Step 3 (a rational box inside a ball). Let $y\in Y$ and let $r>0$ be real. Then there are $a,b\in P$ with $y\in R_{Y}(a,b)\subseteq B_{d}(y,r)$. Let $\delta=r/(4m)$. As $m$ is positive and $0<4$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (vi)), $0<4m$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (iv), so $4m\neq0$ and $\delta=r\cdot(4m)^{-1}$ is a real number (Differences, Quotients and Absolute Values of Real Numbers, and Their Order in the Extended Real Line, Read in the Type Real (ii)); $0<(4m)^{-1}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (viii), applied with $x=y=4m$; and, as $0<r$, $0<\delta$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares (iv). In case V, Rational Tuples and Rational Matrices Approximating a Real Tuple or Matrix from Below and Above in Every Component (i), applied to $y\in\mathbb{R}^{n}$ and $\delta$, and in case M, Rational Tuples and Rational Matrices Approximating a Real Tuple or Matrix from Below and Above in Every Component (ii), applied to $y\in\mathbb{R}^{p\times q}$ and $\delta$, give $a,b\in P$ with $c_{j}(y)-\delta<a_{j}<c_{j}(y)<b_{j}<c_{j}(y)+\delta$ for every $j\in J$, the components of $a$ and $b$ read as real numbers as in the statement; here $c_{j}(y)=y_{j}$ in case V and $c_{(i,l)}(y)=y_{il}$ in case M. So $y\in R_{Y}(a,b)$. Let $z\in R_{Y}(a,b)$ and $j\in J$. Both $c_{j}(y)$ and $c_{j}(z)$ lie strictly between $a_{j}$ and $b_{j}$, and $b_{j}-a_{j}<(c_{j}(y)+\delta)-(c_{j}(y)-\delta)=2\delta$, so $|c_{j}(y)-c_{j}(z)|<2\delta$ (Differences, Quotients and Absolute Values of Real Numbers, and Their Order in the Extended Real Line, Read in the Type Real (iii)). By (E2) with $t=2\delta$, $d(y,z)\le m\cdot(2\delta)=r/2<r$ (The Real Numbers as a Type (vi)), so $z\in B_{d}(y,r)$ by Open Ball in a Metric Space (ii). Hence $R_{Y}(a,b)\subseteq B_{d}(y,r)$.

Proof of (i) and (ii). Let $U$ be open in $(Y,d)$. The map $\rho:P\times P\to\mathcal{P}(Y)$, $(a,b)\mapsto R_{Y}(a,b)$, of the paragraph on the two cases has as image the set $\mathcal{Q}=\{R_{Y}(a,b):a,b\in P\}$; $P\times P$ is countable by Step 1 and Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vii), so $\mathcal{Q}$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vi), and its subset $\mathcal{Q}_{U}=\{W\in\mathcal{Q}:W\subseteq U\}$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (iv). Let $o\in P$ be, in case V, the $n$-tuple all of whose components are $0$ and, in case M, the $p$-tuple all of whose components are the $q$-tuple with all components $0$; then $o_{j}=0$ for every $j\in J$. Since $J$ is nonempty and no real $t$ satisfies $0<t<0$, $R_{Y}(o,o)$ has no element, so $R_{Y}(o,o)=\emptyset\subseteq U$ and $\emptyset\in\mathcal{Q}_{U}$. Thus $\mathcal{Q}_{U}\neq\emptyset$, and by Countable Set (i) there is a surjective map $g:\mathbb{N}\to\mathcal{Q}_{U}$. Let $W_{k}=g(k)$ for $k\in\mathbb{N}$; as $\mathcal{Q}_{U}\subseteq\mathcal{P}(Y)$, $(W_{k})$ is a sequence of subsets of $Y$, and each $W_{k}$ lies in $\mathcal{Q}_{U}$, so $W_{k}\subseteq U$ and $W_{k}=R_{Y}(a,b)$ for some $a,b\in P$. Hence $\bigcup_{k\in\mathbb{N}}W_{k}\subseteq U$, by Indexed Families of Sets and Their Union, Intersection and Product (ii). Conversely, let $y\in U$. As $U$ is open, there is a real $r>0$ with $B_{d}(y,r)\subseteq U$, by Open Subset of a Metric Space (i); Step 3 gives $a,b\in P$ with $y\in R_{Y}(a,b)\subseteq B_{d}(y,r)\subseteq U$. Then $R_{Y}(a,b)\in\mathcal{Q}_{U}$, so $R_{Y}(a,b)=g(k)=W_{k}$ for some $k\in\mathbb{N}$, as $g$ is surjective, and $y\in\bigcup_{k\in\mathbb{N}}W_{k}$. So $U=\bigcup_{k\in\mathbb{N}}W_{k}$. In case V this is (i), and in case M it is (ii).

Step 4 (the coordinate and entry maps are measurable). Let $j\in J$, $y\in Y$ and a real $\varepsilon>0$ be given, and take $\delta=\varepsilon$. If $z\in Y$ and $d(y,z)<\delta$, then, by The Distance on the Real Numbers (i), the symmetry $|u-w|=|w-u|$ of the absolute value (Differences, Quotients and Absolute Values of Real Numbers, and Their Order in the Extended Real Line, Read in the Type Real (iii)) and (E1),

$$d_{\mathbb{R}}(c_{j}(z),c_{j}(y))=|c_{j}(z)-c_{j}(y)|=|c_{j}(y)-c_{j}(z)|\le d(y,z)<\varepsilon .$$

So $c_{j}$ is continuous at every $y\in Y$, that is, continuous, as a map from $(Y,d)$ to $(\mathbb{R},d_{\mathbb{R}})$. The Borel $\sigma$-algebra of $(Y,d)$ is $\mathcal{B}(\mathbb{R}^{n})$ in case V and $\mathcal{B}(\mathbb{R}^{p\times q})$ in case M, and that of $(\mathbb{R},d_{\mathbb{R}})$ is $\mathcal{B}(\mathbb{R})$, by Borel Sigma-Algebra of a Metric Space (iii), Borel Sigma-Algebra of a Metric Space (iv) and Borel Sigma-Algebra of a Metric Space (ii); write $\mathcal{B}(Y)$ for the former. By Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), $c_{j}$ is $\mathcal{B}(Y)/\mathcal{B}(\mathbb{R})$-measurable, so $c_{j}^{-1}(B)\in\mathcal{B}(Y)$ for every $B\in\mathcal{B}(\mathbb{R})$, by Measurable Map, and the Set of Measurable Maps (i).

Proof of (iii) and (iv). Let $\mathcal{C}=\{c_{j}^{-1}(B):j\in J,\ B\in\mathcal{B}(\mathbb{R})\}$, a subset of $\mathcal{P}(Y)$; it is the generating set on the right of (iii) in case V and on the right of (iv) in case M. Let $\mathcal{A}=\sigma(\mathcal{C})$, and let $\mathcal{T}_{d}$ be the set of open subsets of $(Y,d)$, as in Borel Sigma-Algebra of a Metric Space (i), so that $\mathcal{B}(Y)=\sigma(\mathcal{T}_{d})$. By Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), $\mathcal{B}(Y)$ and $\mathcal{A}$ are $\sigma$-algebras on $Y$, and $\mathcal{C}\subseteq\mathcal{A}$.

$\mathcal{A}\subseteq\mathcal{B}(Y)$: by Step 4, $\mathcal{C}\subseteq\mathcal{B}(Y)$, so Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv) applies.

$\mathcal{B}(Y)\subseteq\mathcal{A}$: first let $a,b\in P$. For $j\in J$, $(a_{j},b_{j})\in\mathcal{B}(\mathbb{R})$ by Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii), so $c_{j}^{-1}((a_{j},b_{j}))\in\mathcal{C}\subseteq\mathcal{A}$. As $J$ is countable and nonempty, Unions and Intersections over a Countable Index Set in a σ-Algebra (ii), applied to the map $j\mapsto c_{j}^{-1}((a_{j},b_{j}))$ from $J$ to $\mathcal{A}$, gives $\bigcap_{j\in J}c_{j}^{-1}((a_{j},b_{j}))\in\mathcal{A}$. By the paragraph on the two cases, this intersection is $R_{Y}(a,b)$. Hence $R_{Y}(a,b)\in\mathcal{A}$ for all $a,b\in P$. Now let $U\in\mathcal{T}_{d}$, that is, $U$ open in $(Y,d)$. By (i) in case V and by (ii) in case M, $U=\bigcup_{k\in\mathbb{N}}W_{k}$ for a sequence $(W_{k})$ each of whose terms is $R_{Y}(a,b)$ for some $a,b\in P$; so $(W_{k})$ is a sequence in $\mathcal{A}$ (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v)), and $U\in\mathcal{A}$ by σ-Algebra and Measurable Space (i). Thus $\mathcal{T}_{d}\subseteq\mathcal{A}$, and $\mathcal{B}(Y)=\sigma(\mathcal{T}_{d})\subseteq\mathcal{A}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv).

So $\mathcal{B}(Y)=\mathcal{A}$: in case V this is (iii), and in case M it is (iv).