Compact-open neighbourhoods on the dual give a neighbourhood basis on the group

If $K=\mathrm{\varnothing }$, then $N_{x_1}(K,ϵ)=G$, which is open. Otherwise let $x\in N_{x_1}(K,ϵ)$ and put $\delta :={\max }_{\chi \in K}\mathrm{\mid }\chi (x)-\chi (x_1)\mathrm{\mid }$. The maximum exists because $K$ is nonempty compact and the function is continuous, and $\delta <ϵ$ since every value is strictly below $ϵ$. By [F2] applied with $ϵ-\delta >0$ there is an open neighbourhood $W$ of $0$ with $\mathrm{\mid }\chi (w)-1\mathrm{\mid }<ϵ-\delta$ for all $\chi \in K\cup \{1\}$ and $w\in W$. Then for $w\in W$ and $\chi \in K$, $\mathrm{\mid }\chi (x+w)-\chi (x_1)\mathrm{\mid }\le \mathrm{\mid }\chi (x)\mathrm{\mid }\mathrm{\mid }\chi (w)-1\mathrm{\mid }+\mathrm{\mid }\chi (x)-\chi (x_1)\mathrm{\mid }<ϵ-\delta +\delta =ϵ$, using multiplicativity and $\mathrm{\mid }\chi \mathrm{\mid }\equiv 1$; hence $x+W\subseteq N_{x_1}(K,ϵ)$ and the set is open.

Let $U$ be an open neighbourhood of $0$ in $G$. Choose $g$ as in [F4] and put $f:=g\ast \tilde{g}$. Then $f\in C_c(G)\subseteq L^1(G)\cap L^2(G)$, $\widehat{f}=\mathrm{\mid }\widehat{g}{\mathrm{\mid }}^2\ge 0$ belongs to $L^1(\widehat{G},m_{\widehat{G}})$, and $f(0)=\mathrm{\parallel }g{\mathrm{\parallel }}_2^2>0$; moreover $f(x)={\int }_{\widehat{G}}\widehat{f}(\chi )\chi (x)d{m}_{\widehat{G}}(\chi )$ for every $x\in G$.

Choose a compact $K_0\subseteq \widehat{G}$ with ${\int }_{\widehat{G}\setminus K_0}\widehat{f}<f(0)\mathrm{/}8$ and ${ϵ}_0>0$ with ${ϵ}_0\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_1<f(0)\mathrm{/}4$. For $x\in N_0(K_0,{ϵ}_0)$ we have $\mathrm{\mid }f(x)-f(0)\mathrm{\mid }\le {\int }_{\widehat{G}}\mathrm{\mid }\chi (x)-1\mathrm{\mid }\widehat{f}(\chi )d{m}_{\widehat{G}}(\chi )\le {ϵ}_0\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_1+2{\int }_{\widehat{G}\setminus K_0}\widehat{f}<\frac{f(0)}{2}$, so $f(x)\mathrm{\ne }0$. Since $f$ is continuous and $f(x)\mathrm{\ne }0$ forces $x\in \mathrm{supp}f\subseteq \mathrm{supp}g-\mathrm{supp}g\subseteq U$, we obtain $N_0(K_0,{ϵ}_0)\subseteq U$.

For every $x_1\in G$ and every neighbourhood $V$ of $x_1$, choose an open neighbourhood $V_0$ of $x_1$ contained in $V$ and apply step 2.1 to the open identity neighbourhood $U:=V_0-x_1$. This produces compact $K_0$ and ${ϵ}_0>0$ with $x_1+N_0(K_0,{ϵ}_0)=N_{x_1}(K_0,{ϵ}_0)\subseteq V$; since $x_1+N_0(K_0,{ϵ}_0)$ is a neighbourhood of $x_1$, the sets $N_{x_1}(K,ϵ)$ form a neighbourhood basis at $x_1$, each of them open by step 1.1.

Equip $\mathrm{\Phi }(G)$ with the subspace topology from $\widehat{\widehat{G}}$. For compact $L\subseteq \widehat{G}$ and open $V\subseteq \mathbb{T}$, if $L=\mathrm{\varnothing }$ the preimage under $\mathrm{\Phi }$ of $\{\xi :\xi [L]\subseteq V\}$ is all of $G$. Otherwise, let $x_0$ lie in that preimage. Joint continuity of evaluation [F1] gives, for each $\chi \in L$, open neighbourhoods $U_{\chi }\ni x_0$ and $W_{\chi }\ni \chi$ with $\eta (x)\in V$ for $(x,\eta )\in U_{\chi }\times W_{\chi }$. Compactness of $L$ supplies a finite subcover $W_{{\chi }_1},\dots ,W_{{\chi }_m}$; then $U:={\bigcap }_{j=1}^m{U}_{{\chi }_j}$ is a neighbourhood of $x_0$ contained in that preimage. Thus $\mathrm{\Phi }$ is continuous. Conversely, fix $x_1\in G$, compact $K\subseteq \widehat{G}$ and $ϵ>0$. If $K=\mathrm{\varnothing }$, then $N_{x_1}(K,ϵ)=G$ and its image is open in $\mathrm{\Phi }(G)$. Otherwise set $F(\xi ,\chi ):=\mathrm{\mid }\xi (\chi )-\chi (x_1)\mathrm{\mid }$ on $\widehat{\widehat{G}}\times K$. This is continuous by the joint evaluation pairing and continuity of the fixed evaluation at $x_1$ [F1], and $F(\mathrm{\Phi }(x_1),\chi )=0$ for every $\chi \in K$. For each $\chi \in K$, continuity gives open neighbourhoods $A_{\chi }\ni \mathrm{\Phi }(x_1)$ and $B_{\chi }\ni \chi$ on which $F<ϵ$; choose a finite subcover $B_{{\chi }_1},\dots ,B_{{\chi }_m}$ of $K$. Then $A:={\bigcap }_{j=1}^m{A}_{{\chi }_j}$ is open and lies in $\{\xi :\mathrm{\mid }\xi (\chi )-\chi (x_1)\mathrm{\mid }<ϵ\text{ for all }\chi \in K\}$. Thus $A\cap \mathrm{\Phi }(G)\subseteq \mathrm{\Phi }(N_{x_1}(K,ϵ))$. For any open $U\subseteq G$ and each $x_1\in U$, step 3.1 supplies one such basis set contained in $U$; the corresponding $A$ is an open neighbourhood of $\mathrm{\Phi }(x_1)$ whose trace lies in $\mathrm{\Phi }(U)$. Therefore $\mathrm{\Phi }(U)$ is open in $\mathrm{\Phi }(G)$, and $\mathrm{\Phi }$ is open onto its image. By [F7] the map $\mathrm{\Phi }$ is injective, hence a homeomorphism onto its image; by step 3.1 the topology of $G$ is the topology of uniform convergence on compact subsets of $\widehat{G}$ transported by $\mathrm{\Phi }$.