Corollary 2

Let H be a closed subgroup of G.

a) In order that H be compact, it is necessary and sufficient that $H^{\bot}$ be open in $\widehat{G}$ ;

b) In order that H be open, it is necessary and sufficient that $H^{\bot}$ be compact in $\widehat{G}$.

a) To say that $H^{\bot}$ is open amounts to saying that $\widehat{G}/H^{\bot}$ is discrete, now $\widehat{G}/H^{\bot}$ is isomorphic to $\widehat{H}$ (Theorem 4 of II, p. 226); the assertion therefore follows from Proposition 18. Assertion b) results by duality from assertion a) applied to $H^{\bot}$.