Topological groups
Introduction
A topological group is a mathematical structure that combines the properties of a group and a topological space in a compatible way. This compatibility is expressed through the continuity of the group operations, which allows for a rich interplay between algebraic and topological properties.
Definition
Formally, a topological group is a group G equipped with a topology such that the group operations of multiplication and taking inverses are continuous functions. In more detail, if we denote the group multiplication by ⋅: G × G → G and the function taking each element to its inverse by −1: G → G, then these functions are required to be continuous when G × G and G are equipped with the product topology and the given topology, respectively.
Examples
There are many examples of topological groups. Perhaps the simplest examples are the real numbers R with the standard topology and the usual addition operation, and the circle group T = {e^iθ : θ ∈ R}, which is a subgroup of the complex numbers under multiplication.
Properties
Topological groups have many interesting and useful properties. For instance, every topological group is a uniform space in a natural way, and therefore a completely regular topological space. Moreover, the concepts of compactness, connectedness, and completeness can all be defined for topological groups.
Subgroups and Quotient Groups
A subgroup of a topological group that is also a topological group (with the subspace topology) is called a topological subgroup. Similarly, if G is a topological group and H is a normal subgroup of G, then the quotient group G/H can be given a topology such that it becomes a topological group.
Homomorphisms and Isomorphisms
A homomorphism between two topological groups that is also a continuous function is called a continuous homomorphism. If a continuous homomorphism is also a homeomorphism, it is called a topological isomorphism, and the groups are said to be isomorphic as topological groups.
Topological Group Actions
A topological group action is an action of a topological group on a topological space that is continuous in the sense that for any point in the space, its orbit under the action is a continuous image of the group.
Applications
Topological groups have many applications in various areas of mathematics, including algebra, analysis, geometry, and number theory. For instance, they are fundamental in the theory of Lie groups, which play a central role in many parts of mathematics and theoretical physics.
