Open subgroup implies closed

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This article gives the statement and possibly, proof, of an implication relation between two topological subgroup properties. That is, it states that every subgroup of a topological group satisfying the first subgroup property must also satisfy the second
View a complete list of topological subgroup property implications

Statement

For semitopological groups

Any open subgroup of a left-topological group or right-topological group is closed.

Any 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.

Related facts

Corollaries

Similar facts

Facts used

  1. Left cosets partition a group

Proof

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

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

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

To prove: H is a closed 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 an open subset of G. Homeomorphisms take open subsets to open subsets H is open in G Step (1) [SHOW MORE]
3 The union of all the left cosets of H other than H itself is open in G Union of open subsets is open Step (2) Step-fact combination direct.
4 H is closed in G A subset is closed iff its set-theoretic complement is open. Also, Fact (1) Step (3) [SHOW MORE]

Proof for right-topological group

The proof is analogous, but we use right cosets instead of left cosets.