subgroup of topological group is either clopen or has empty interior
Theorem - Every subgroup of a topological group is either clopen or has empty interior.
Proof: Let be a topological group and a subgroup. Suppose the interior of is nonempty, i.e. there is a non-empty open set of such that . Translating around we can see that is open: if then for every the set is open in , is contained in and contains , which implies that is open in .
Let us now see that is closed. Let denote the closure of and let be the set of elements of the form where . Of course, since is a subgroup of , we have that . Also, since is open we know that (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups) - 5). Hence , i.e. is closed.
We have proven that a subgroup of a topological group must be clopen or it must have empty interior. Since this two topological properties can never be satisfied simultaneously, we have that every subgroup of a topological group is either clopen or it has empty interior.
|Title||subgroup of topological group is either clopen or has empty interior|
|Date of creation||2013-03-22 18:01:29|
|Last modified on||2013-03-22 18:01:29|
|Last modified by||asteroid (17536)|