connected topological group is generated by any neighborhood of identity
Proof: Let be an open neighborhood of . For each we denote by the set of elements of the form , where each . Let .
Since each is open (see this entry (http://planetmath.org/BasicResultsInTopologicalGroups) - 3), we have that is an open set. We now see that it is also closed.
Let , the closure of . Since is an open neighborhood of , it must intersect . Thus, let .
Since , then for some element .
Since , then for some , i.e. with each .
We then have , i.e. . Hence, is closed.
Since is connected and is open and closed, we must have . This means that is generated by .
|Title||connected topological group is generated by any neighborhood of identity|
|Date of creation||2013-03-22 18:01:45|
|Last modified on||2013-03-22 18:01:45|
|Last modified by||asteroid (17536)|