# Closed subgroup of finite index implies open

## Statement

### Statement for left-topological, right-topological, or semitopological groups

In a left-topological group or right-topological group, any closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index) must be an 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 closed subgroup of finite index (i.e., a closed subgroup that is also a subgroup of finite index) must be an open subgroup.

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

## Proof

### 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:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 For all $g \in G$, the map $G \to G$ given by $x \mapsto gx$ is a self-homeomorphism of $G$. Definition of left-topological group $G$ is a left-topological group. [SHOW MORE]
2 Every left coset of $H$ in $G$ is a closed subset of $G$. Homeomorphisms take closed subsets to closed subsets Step (1) By Step (1), $x \mapsto gx$ is a self-homeomorphism of $G$, so it takes the closed subset $H$ to the closed subset $gH$. Thus, for any $g \in G$, $gH$ is closed in $G$.
3 The union of all the left cosets of $H$ other than $H$ itself is closed in $G$ Union of finitely many closed subsets is closed $H$ has finite index in $G$ Step (2) Step-fact combination direct.
4 $H$ is open in $G$ A subset is open iff its set-theoretic complement is closed. Step (3) The set-theoretic complement of $H$ in $G$ is precisely the union of all the left cosets other than $H$ itself, and by Step (3), this is closed. Hence, $H$ is open.

### 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.