Let $G$ be an abelian topological group, $G^\vee:=\{f\in\text{Hom}(G,\mathbb{T})\text{ continuous}\}$ its Pontryagin dual and $\widehat{G}:=\{\pi:G\to\text{U}(\mathcal{H})\text{ irreducible unitary representation}\}/_\cong$ its unitary dual. Using Schur's lemma, one can easily see that sending a unirrep $\pi:G\to\text{U}(\mathcal{H})$ to the character $G\to\mathbb{T}$, $g\mapsto\lambda_g$, where $\pi(g)=\lambda_g\text{id}_\mathcal{H}$, is a bijection $\varphi:\widehat{G}\to G^\vee$.
For Pontryagin duality to work, one equips $G^\vee$ with the compact-open topology and for Kirillov's orbit method to work, one equips $\widehat{G}$ with the Fell topology. Is $\varphi$ a homeomorphism then? If not, is it at least true if we restrict ourselves to locally compact groups or just $\mathbb{R}^n$? (For the case of $\mathbb{R}^n$ I think the continuity of $\varphi^{-1}$ is easy to see, since an isomorphism $\mathbb{R}^n\to(\mathbb{R}^n)^\vee$ is given by the exponential map.)
The definitions of the compact-open topology and the Fell topology seem to be quite similar, but I don't know how a proof of them being homeomorphic might look like. Any help or reference is very much appreciated!
Edit: The Fell topology on $\widehat{G}$ is generated by sets of the form $U_{K,\epsilon,x_1,...,x_n}(\pi):=\{\rho:G\to\text{U}(\mathcal{V})\ |\ \exists y_1,...,y_n\in\mathcal{V}:\ |\langle\pi(g)(x_i),x_j\rangle-\langle\rho(g)(y_i),y_j\rangle|<\epsilon\ \forall g\in K\ i,j=1,...,n\}$, where $K\subset G$ is compact, $\epsilon>0$, $x_1,...,x_n\in\mathcal{H}$ and $\pi$ as above.