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Let $G$ be a compact topological group (we don't assume any separation axiom here). For a subgroup $H$ of $G$, we may endow the quotient space $G/H$ with quotient topology. Now it is clear for me that openness of $H$ implies finiteness of $G/H$. My question is whether the converse is also true.

If not, I would like to hear some counterexamples in the case of Lie groups and profinite groups.

Rigid AOE2
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    Counterexamples in terms of profinite groups can be found here: https://math.stackexchange.com/questions/1128362/if-g-is-a-compact-topological-group-how-to-show-that-a-finite-index-subgroup – Rigid AOE2 Nov 20 '20 at 07:22
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    Counterexamples in terms of Lie groups don't exist: https://math.stackexchange.com/questions/454320/when-are-finite-index-subgroups-of-a-lie-group-closed – Rigid AOE2 Nov 20 '20 at 07:23

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