Mathematics > Topology > Topological Groups
Description:
Topology is a branch of mathematics that deals with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Within this broad field, the study of topological groups is a specialized area that focuses on the interplay between algebraic and topological structures.
A topological group is a mathematical structure that merges two fundamental concepts: a group in algebra and a topology in the context of a geometric or analytical structure. Specifically, a topological group \( G \) is a group that is also a topological space, such that the group operations (multiplication and inversion) are continuous with respect to the topology on \( G \).
Definition and Properties:
Formally, let \( (G, \cdot) \) be a group and let \( (G, \tau) \) be a topological space. The pair \( (G, \tau, \cdot) \) is called a topological group if the following two conditions are satisfied:
Continuity of the Group Operation:
The group multiplication \( m: G \times G \to G \), defined by \( m(x, y) = x \cdot y \) for all \( x, y \in G \), is continuous. This means that for any open set \( U \subset G \), the preimage \( m^{-1}(U) \) is an open set in the product topology of \( G \times G \).\[
m^{-1}(U) \in \tau \times \tau \quad \forall U \in \tau
\]Continuity of the Inverse Operation:
The inversion map \( i: G \to G \), defined by \( i(x) = x^{-1} \) for all \( x \in G \), is continuous. This ensures that for any open set \( V \subset G \), the preimage \( i^{-1}(V) \) is an open set in \( \tau \).\[
i^{-1}(V) \in \tau \quad \forall V \in \tau
\]
Examples:
The Real Numbers \( \mathbb{R} \) with Addition:
The set of real numbers \( \mathbb{R} \) under addition forms a topological group when endowed with the standard topology of the real line. The addition operation \( (x, y) \mapsto x + y \) and the negation operation \( x \mapsto -x \) are both continuous functions in this context.The Circle Group \( \mathbb{T} \):
The unit circle \( \mathbb{T} = \{ z \in \mathbb{C} \mid |z| = 1 \} \) in the complex plane forms a topological group under multiplication of complex numbers. The standard topology on \( \mathbb{T} \) makes both the multiplication \( z_1 \cdot z_2 \) and the inversion \( z^{-1} \) operations continuous.
Fundamental Concepts:
Homomorphism: A continuous group homomorphism between two topological groups \( G \) and \( H \) is a function \( \phi: G \to H \) that respects both the group structure and the topological structure. That is, \( \phi(x \cdot y) = \phi(x) \cdot \phi(y) \) and \( \phi \) is continuous.
Quotient Topological Groups: Given a topological group \( G \) and a closed normal subgroup \( N \), the quotient group \( G/N \) can be given a natural topology, known as the quotient topology, making \( G/N \) into a topological group.
Compact and Locally Compact Groups: A topological group \( G \) is compact if every open cover of \( G \) has a finite subcover, and it is locally compact if every point has a compact neighborhood. Examples include finite groups under the discrete topology and the torus \( \mathbb{T}^n \).
Applications:
Topological groups play a crucial role in various fields such as harmonic analysis, where they provide a natural setting for Fourier analysis, and in theoretical physics, particularly in the study of symmetry and group actions in quantum mechanics and relativity.
By examining structures that embody both algebraic and topological properties, mathematics gains powerful tools for addressing complex problems that arise in diverse branches of science and engineering. The theory of topological groups thus stands as a vibrant intersection of disciplines, offering deep insights and broad applications.