Dual homomorphisms: continuity, and the annihilator of a closed subgroup

Part (a): for $\gamma \in \widehat{H}$ the composite $\gamma \circ \phi$ is continuous by [F3] and is a homomorphism, so $\widehat{\phi }(\gamma )\in \widehat{G}$; $\widehat{\phi }$ is a group homomorphism because $\widehat{\phi }({\gamma }_1{\gamma }_2)(x)={\gamma }_1(\phi (x)){\gamma }_2(\phi (x))=(\widehat{\phi }{\gamma }_1)(x)(\widehat{\phi }{\gamma }_2)(x)$. For compact $K\subseteq G$ and open $V\subseteq \mathbb{T}$ the preimage satisfies ${\widehat{\phi }}^{-1}(S_G(K,V))=\{\gamma \in \widehat{H}:\gamma [\phi [K]]\subseteq V\}=S_H(\phi [K],V)$, and $\phi [K]$ is compact by [F3]; this set is subbasic open in $\widehat{H}$ by [F1], so $\widehat{\phi }$ is continuous. Pullback preserves the identity map and reverses composition: for $\psi :H\to J$, associativity gives $\gamma \circ (\psi \circ \phi )=(\gamma \circ \psi )\circ \phi$, hence $\widehat{\psi \circ \phi }=\widehat{\phi }\circ \widehat{\psi }$.

Part (b), quotient topology. For every open $O\subseteq G$, $q^{-1}(q(O))=O+H={\bigcup }_{h\in H}(O+h)$ is open by translations, hence $q(O)$ is open by the quotient topology. Thus $q$ is open. Distinct cosets $q(x),q(y)$ have $x-y\mathrm{\notin }H$. Closedness of $H$ gives an open neighbourhood $W$ of $x-y$ disjoint from $H$. By continuity of subtraction choose identity neighbourhoods $U,V$ with $(x-y)+U-V\subseteq W$. The open sets $q(x+U)$ and $q(y+V)$ are disjoint: an intersection would give $x+u-y-v\in H\cap W$. Therefore $G\mathrm{/}H$ is Hausdorff. For any $x\in G$, choose a compact neighbourhood $N$ of $x$ and open $U$ with $x\in U\subseteq N$. Then $q(N)$ is compact by continuity, closed because the quotient is Hausdorff, and contains the open neighbourhood $q(U)$ of $q(x)$. Thus the quotient is locally compact. The product $q\times q$ is a continuous open surjection: images of basic open rectangles are open rectangles, and arbitrary opens are unions of those rectangles. It is therefore quotient by [F6]. The quotient multiplication is continuous since its composite with $q\times q$ is the continuous map $q\circ m_G$, and the quotient universal property [F4] applies; similarly inversion descends through $q$. Hence $G\mathrm{/}H$ is an abelian topological group.

$H^{\perp }$ is closed in $\widehat{G}$: it is the intersection over $h\in H$ of the sets $\{\gamma :\gamma (h)=1\}$, each of which is the preimage of the closed set $\{1\}\subseteq \mathbb{T}$ under the evaluation map $\gamma \mapsto \gamma (h)$, and that evaluation is continuous because $\{\gamma :\gamma (h)\in V\}=S(\{h\},V)$ is subbasic open for every open $V\subseteq \mathbb{T}$.

Compact lifts, proved locally. Let $L\subseteq G\mathrm{/}H$ be compact. If $L=\mathrm{\varnothing }$, take $K=\mathrm{\varnothing }$. Otherwise, for every $l\in L$ the set of triples $(x,N,U)$ with $q(x)=l$, $N$ a compact neighbourhood of $x$, and $U$ open with $x\in U\subseteq N$ is nonempty by surjectivity and [F8]. Use [A1] to choose such a triple $(x_l,N_l,U_l)$ for each $l$; this is the only invocation of Choice in this argument. By step 1.2 the sets $q(U_l)$ form an open cover of $L$. The indexed ambient-cover criterion [F9] gives finitely many indices whose sets cover $L$. Take their associated triples, and let $K$ be the union of the corresponding $N_l$. Finite unions of compact subsets are compact by [F8], and $L\subseteq {\bigcup }_{l}q(U_l)\subseteq q(K)$. This proves the compact-lift theorem needed below from earlier topology alone.

The pullback $\widehat{q}:\widehat{G\mathrm{/}H}\to \widehat{G}$ is a continuous group homomorphism by step 1.1 applied to the continuous homomorphism $q$; its image lies in $H^{\perp }$ because $\widehat{q}(\gamma )(h)=\gamma (q(h))=\gamma (0)=1$ for $h\in H$, and $\widehat{q}$ is injective because $\gamma \circ q=1$ forces $\gamma =1$, the quotient map $q$ being surjective.

The image of $\widehat{q}$ equals $H^{\perp }$: if $\gamma \in \widehat{G}$ satisfies $\gamma [H]=\{1\}$, then $\gamma$ is constant on the fibres of $q$, for $q(x)=q(x^{\mathrm{\prime }})$ means $x-x^{\mathrm{\prime }}\in H$ and then $\gamma (x)=\gamma (x^{\mathrm{\prime }})\gamma (x-x^{\mathrm{\prime }})=\gamma (x^{\mathrm{\prime }})$; the quotient universal property [F4] factors $\gamma ={\gamma }^{\mathrm{\prime }}\circ q$ with ${\gamma }^{\mathrm{\prime }}$ continuous, and ${\gamma }^{\mathrm{\prime }}$ is a homomorphism because for cosets $a=q(x)$, $b=q(y)$ one has ${\gamma }^{\mathrm{\prime }}(a+b)={\gamma }^{\mathrm{\prime }}(q(x+y))=\gamma (x+y)=\gamma (x)\gamma (y)={\gamma }^{\mathrm{\prime }}(a){\gamma }^{\mathrm{\prime }}(b)$. Hence $\gamma =\widehat{q}({\gamma }^{\mathrm{\prime }})\in \widehat{q}[\widehat{G\mathrm{/}H}]$.

The inverse ${\widehat{q}}^{-1}:H^{\perp }\to \widehat{G\mathrm{/}H}$ is continuous: fix ${\gamma }_0\in H^{\perp }$ and put ${\delta }_0:={\widehat{q}}^{-1}({\gamma }_0)$, and let $S(L,V)=\{\delta :\delta [L]\subseteq V\}$ be a subbasic open set containing ${\delta }_0$, with $L\subseteq G\mathrm{/}H$ compact and $V\subseteq \mathbb{T}$ open. Since $L$ is compact and ${\delta }_0$ is continuous, ${\delta }_0[L]$ is compact by [F3]; consider all pairs $(A,B)$ of open subsets of $\mathbb{T}$ with $1\in B$ and $AB\subseteq V$. The first coordinates $A$ of these pairs cover ${\delta }_0[L]$: for every $t\in {\delta }_0[L]\subseteq V$, continuity of multiplication at $(t,1)$ provides such a pair with $t\in A$. The ambient-cover criterion [F9] selects finitely many first coordinates $A_1,\dots ,A_n$ covering ${\delta }_0[L]$, and finite choice [F9] selects their associated $B_i$. Set $W={\bigcap }_{i=1}^n{B}_i$, an open neighbourhood of $1$ with ${\delta }_0(l)W\subseteq V$ for every $l\in L$; for empty $L$, use $W=\mathbb{T}$. This construction uses no arbitrary-index choice. By the compact-lift theorem in step 2.1 choose compact $K\subseteq G$ with $L\subseteq q[K]$; then $N:=({\gamma }_0\cdot S_G(K,W))\cap H^{\perp }$ is a neighbourhood of ${\gamma }_0$ in $H^{\perp }$, because $S_G(K,W)$ is a subbasic open neighbourhood of $1$ in $\widehat{G}$ and translation by ${\gamma }_0$ is a homeomorphism by [F2]. For $\gamma ={\gamma }_0\eta \in N$ with $\eta \in S_G(K,W)$ and $l=q(k)\in L$ with $k\in K$ one has ${\widehat{q}}^{-1}(\gamma )(l)=\gamma (k)={\gamma }_0(k)\eta (k)={\delta }_0(l)\eta (k)\in {\delta }_0(l)W\subseteq V$; hence ${\widehat{q}}^{-1}[N]\subseteq S(L,V)$ and ${\widehat{q}}^{-1}$ is continuous at the arbitrary point ${\gamma }_0$.

Conclusion of (b): by steps 2.2, 3.1 and 4.1 the map $\widehat{q}$ is a group isomorphism of $\widehat{G\mathrm{/}H}$ onto $H^{\perp }$ that is continuous and has continuous inverse, hence a topological group isomorphism onto $H^{\perp }$; $H^{\perp }$ is closed in $\widehat{G}$ by step 1.3, and it is a subgroup because it is the kernel of the homomorphism $\gamma \mapsto (\gamma (h){)}_{h\in H}$ restricted to the abelian group $\widehat{G}$. Together with part (a), proved in step 1.1, this is the statement.