Open subgroup implies closed

For semitopological groups
Any uses property satisfaction of::open subgroup of a  left-topological group or  right-topological group is  closed.

Any uses property satisfaction of::open subgroup of a  semitopological group is  closed.

Since any topological group is a semitopological group (see topological group implies semitopological group), this in particular tells us that any open subgroup of a topological group is closed.

For other types of groups
The statement is true for algebraic groups as well as for Lie groups, where open and closed are interpreted in terms of the corresponding topologies. This is because algebraic groups are in particular semitopological groups (though they are not topological groups) and Lie groups are in particular topological groups with the corresponding topologies.

Corollaries

 * Connected implies no proper open subgroup

Similar facts

 * Closed and finite index implies open

Facts used

 * 1) uses::Left cosets partition a group

Proof outline
The idea behind the proof is to show that if the subgroup is open, i.e., all its points are well inside it, then each of its cosets is open, i.e., all points outside it are well outside it. This shows that the subgroup is closed. We use left cosets for left-topological groups and right cosets for right-topological groups.

Proof for left-topological group
Given: A left-topological group $$G$$, an open subgroup $$H$$ of $$G$$.

To prove: $$H$$ is a closed subgroup of $$G$$

Proof:

Proof for right-topological group
The proof is analogous, but we use right cosets instead of left cosets.