basic results in topological groups
The purpose of this entry is to list some and useful results concerning the topological of topological groups. We will use the following notation whenever are subsets of a topological group and an element of :
denotes the closure of
2 - Let be a topological group and the identity element. Let be a neighborhood base around . Then is a neighborhood base around and is a basis (http://planetmath.org/BasisTopologicalSpace) for the topology of .
3 - Let be a topological group. If is open and is any subset of , then is an open set in .
5 - Let be a topological group and the identity element. If is a neighborhood of then .
6 - Let be a topological group, the identity element and a neighborhood around . Then there exists a neighborhood around such that .
7 - Let be a topological group, the identity element and a neighborhood around . Then there exists a symmetric (http://planetmath.org/SymmetricSet) neighborhood around such that .
8 - Let be a topological group. If is a subgroup of , then so is .
9- Let be a topological group. If is an open subgroup of , then is also closed.
|Title||basic results in topological groups|
|Date of creation||2013-03-22 17:37:38|
|Last modified on||2013-03-22 17:37:38|
|Last modified by||asteroid (17536)|