Corollary 3

Let $(H_i)_{i\in I}$ be a decreasing filtered family of compact subgroups of G. In order that G be identified with the projective limit of the groups $G/H_i$, it is necessary and sufficient that $\widehat{G}$ be the union of the open subgroups $H^{\bot}_i$.

To say that G is identified with the projective limit of the $G/H_i$ amounts to saying that $\bigcap_iH_i=\{e\}$ (TG, III, p. 60, Proposition 2), that is, that $\bigcup_iH^{\bot}_i$ is dense in $\widehat{G}$ (Corollary 4 of Theorem 4 of II, p. 226). Now $\bigcup_iH^{\bot}_i$ is an open subgroup, hence closed, of $\widehat{G}$.