Annihilators of closed subgroups of Euclidean space

Given: The group ${\mathbb{R}}^n$ with its standard topology and the dual identification constructed from the character classification of the line and the product-dual identification (Continuous characters of the real line are exponentials, Duals of finite products and of discrete direct sums).

Every continuous homomorphism $\phi :{\mathbb{R}}^n\to \mathbb{T}$ is $\phi (x)=e^{2\pi i\xi \cdot x}$ for a unique $\xi \in {\mathbb{R}}^n$, and $\xi \mapsto {\phi }_{\xi }$ is an isomorphism of topological groups ${\mathbb{R}}^n\to \widehat{{\mathbb{R}}^n}$: the classification of characters of the line supplies the algebraic bijection, and finite-product duality reduces the topology check to the line. On the line, compact $K$ is bounded, say $\mathrm{\mid }t\mathrm{\mid }\le M$; continuity of $s\mapsto e^{2\pi is}$ at $0$ makes $e^{2\pi i\xi t}$ uniformly close to $e^{2\pi i{\xi }_{0}t}$ on $K$ when $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }$ is small. Conversely, if $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }\ge ϵ$, then $t=1\mathrm{/}(2\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid })\in [-1\mathrm{/}ϵ,1\mathrm{/}ϵ]$ gives $\mathrm{\mid }{e}^{2\pi i(\xi -{\xi }_0)t}-1\mathrm{\mid }=2$, so the uniform ball of radius $1$ on this compact interval forces $\mathrm{\mid }\xi -{\xi }_0\mathrm{\mid }<ϵ$. Uniform balls are compact-open neighbourhoods by The compact-open character group is a Hausdorff topological abelian group, and compactness and boundedness of these line sets follow from Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. The value at $\pm \pi i$ is $-1$ by $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. At $n=0$ both groups are trivial. (Continuous characters of the real line are exponentials, Duals of finite products and of discrete direct sums, The Pontryagin dual with the compact-open topology)

$H^{\perp }=\{\gamma :\gamma (h)=1\text{ for every }h\in H\}$, and $e^{2\pi it}=1$ exactly when $t\in \mathbb{Z}$, while $e^{2\pi i(t+k)}=e^{2\pi it}$ for $k\in \mathbb{Z}$. (The annihilator of a subgroup, $\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$, The integers as equivalence classes of pairs of naturals)

The quotient ${\mathbb{R}}^n\mathrm{/}H$ for $H={\mathbb{Z}}^k\times \{0{\}}^{n-k}$ is $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^k\times {\mathbb{R}}^{n-k}$ with the product topology: each projection $\mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}$ is continuous and open, since the saturation of an open set is the union of its integer translates. The finite product of these maps and identity maps is therefore continuous, open and surjective, with kernel $H$, and the induced quotient bijection is continuous and open. A continuous character on ${\mathbb{R}}^n$ constant on the cosets of $H$ factors through the quotient map, uniquely and continuously; conversely characters of the quotient pull back to characters of ${\mathbb{R}}^n$ that are trivial on $H$. (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)

For a real matrix $A$, a vector $\xi$ and $z\in {\mathbb{Z}}^n$ one has $(A^T\xi )\cdot z=\xi \cdot (Az)$ and $A^{-T}=(A^T{)}^{-1}$; the standard basis vectors $e_0,\dots ,e_{n-1}$ lie in ${\mathbb{Z}}^n$, and $y\in {\mathbb{R}}^n$ satisfies $y\cdot z\in \mathbb{Z}$ for all $z\in {\mathbb{Z}}^n$ exactly when $y\in {\mathbb{Z}}^n$. (Transpose is linear and involutive, and $(AB{)}^{\mathsf{T}}=B^{\mathsf{T}}{A}^{\mathsf{T}}$, The transpose $A^{\mathsf{T}}$ of a matrix, Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, The integers as equivalence classes of pairs of naturals)