Open subgroup implies closed
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
For semitopological groups
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.
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
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Given: A left-topological group , an open subgroup of .
To prove: is a closed subgroup of
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||For all , the map given by is a self-homeomorphism of .||Definition of left-topological group||is a left-topological group.||[SHOW MORE]|
|2||Every left coset of in is an open subset of .||Homeomorphisms take open subsets to open subsets||is open in||Step (1)||[SHOW MORE]|
|3||The union of all the left cosets of other than itself is open in||Union of open subsets is open||Step (2)||Step-fact combination direct.|
|4||is closed in||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.