Topological group. From a logical perspective, it results as the fusion of the concepts of group and topological space . In the same set H, the group multiplication and topological adherence operations are defined together. However, these operations are linked by the continuity requirement: the group operations that exist in H must be continuous in the topological space H.
History
In the historical succession of ideas, the concept of topological group appeared linked to the study of groups of continuous transformations. When in Euclidean space , for example, a group of continuous applications are established , in this same group, limit relations emerge naturally. This group becomes a topological group. The advance of the investigations located that the most important properties considered are based, clearly, on the limit relations that exist in the same group, regardless of the case that the group considered is a group of transformations.
Definition
A set H is called a topological group if
- H is a group, this is an algebraic system with a law of internal composition.
- H is a topological space.
- The group operations defined in H are continuous in the topological space H. More formally, this formality can be expressed, as follows:
- If c and d are elements of the set H, for every neighborhood W of element ab there are neighborhoods U and V of a and b respectively, such that UV ⊂ W.
- If c is an element of the set H, for every neighborhood V of of the element c -1 , there is a neighborhood U of the element c, which U -1 ⊂ V. However, the two previous conditions can be synthesized in a unique way:
- If c and d are two elements of the set H, for every neighborhood W of the element cd -1 , there are neighborhoods of the elements c and d such that UV -1 ⊂ W
