Let $G$ be a locally compact Hausdorff abelian group, and $H$ a closed subgroup of $G$. Let $\hat{G}$ denote the Pontraygin dual of $G$, i.e. the group of coninuous homomorphisms $G \rightarrow S^1$ in the compact-open topology. The notes I'm reading claims there is an isomorphism of topological groups $\hat{G} / H^{ \perp} \cong \hat{H}$, where $H^{\perp}$ denotes the subgroup of $\chi \in \hat{G}$ for which $\chi(H) = \{1\}$.
From the inclusion map $\phi: H \rightarrow G$, I know that we get a corresponding continuous map $\hat{\phi}: \hat{G} \rightarrow \hat{H}$ given by $\hat{\phi}(\chi) = \chi \circ \phi = \chi |_H$. The kernel of this map is $H^{\perp}$, so $H^{\perp}$ is a closed subgroup of $\hat{G}$, and this induces a continuous injective map $$\overline{\hat{\phi}}: \hat{G}/H^{\perp} \rightarrow \hat{H}$$ But why is this map surjective? In other words, why is every character on $H$ the restriction of a character on $G$? Also, this map $\overline{\hat{\phi}}$ is continuous, but in order for there to be an isomorphism of topological groups, it would need to be bicontinuous (an open map). Why is this?
