Closed subgroup of finite index implies open

Statement for left-topological, right-topological, or semitopological groups
In a left-topological group or right-topological group, any uses property satisfaction of::closed subgroup of finite index (i.e., a  uses property satisfaction of::closed subgroup that is also a  uses property satisfaction of::subgroup of finite index) must be an  proves property satisfaction of::open subgroup.

Note that a semitopological group is both a left-topological group and a right-topological group, so the result applies to semitopological groups.

Statement for topological groups
In a topological group, any uses property satisfaction of::closed subgroup of finite index (i.e., a  uses property satisfaction of::closed subgroup that is also a  uses property satisfaction of::subgroup of finite index) must be an  proves property satisfaction of::open subgroup.

Note that topological groups are semitopological groups, so the result applies to these.

Related facts

 * Open subgroup implies closed
 * Connected implies no proper open subgroup
 * Compact implies every open subgroup has finite index

Proof for left-topological groups
Given: A right-topological group $$G$$, a closed subgroup $$H$$ of finite index in $$G$$.

To prove: $$H$$ is an open subgroup of $$G$$

Proof:

Proof for right-topological groups
The proof is analogous to the proof for left-topological groups, except that we use right cosets instead of left cosets.