"In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time."
A topological group is a group equipped with a topology that is compatible with the group operations. This concept is frequently used in Lie theory, functional analysis, and geometry.
Topological spaces: A topological space is a set with a topology, which describes the properties of the set and its subsets with respect to open sets.
Group theory: Group theory is the study of symmetry and transformation in mathematical structures. It deals with the properties of groups, which are sets with an operation that satisfies certain axioms.
Continuous functions: A function is said to be continuous if small changes in the input lead to small changes in the output. In topology, continuity is defined in terms of open sets.
Convergence: Convergence is the idea that a sequence of points in a topological space approaches a limit. This idea is central to much of topology and analysis.
Hausdorff spaces: A Hausdorff space is a topological space where any two distinct points can be separated by disjoint open sets.
Compactness: A topological space is said to be compact if every open cover has a finite subcover. Compact spaces have many useful properties.
Connectedness: A topological space is connected if it cannot be split into two disjoint open sets. Connected spaces have important properties.
Quotient Spaces: A quotient space is a space constructed from another space by identifying certain points.
Homotopy equivalence: Two spaces are homotopy equivalent if they can be continuously deformed into each other. Homotopy equivalence is a fundamental concept in algebraic topology.
Cohomology: Cohomology is a branch of algebraic topology that measures the extent to which topological spaces fail to be globally contractible.
Due to the vast number of possible topological groups, it is not possible to provide an exhaustive list along with a brief description of each type: However, some examples of topological groups and their properties are:.
Discrete topological groups: A topological group where every point is an isolated point, i.e., for any point x in the group, there exists an open set containing x that does not contain any other point of the group. Examples include the integers with the discrete topology, any finite group with the discrete topology, and the product of discrete topological groups.
Compact topological groups: A topological group that is also a compact space, i.e., every open cover has a finite subcover. Examples include the circle group (the group of complex numbers on the unit circle), the torus (the product of two circles), and the orthogonal group (the group of linear isometries of a Euclidean space).
Locally compact topological groups: A topological group that has a compact open neighborhood around every point. Examples include the real line with the usual topology and the invertible elements of any Banach algebra with the norm topology.
Hausdorff topological groups: A topological group where every two distinct points have disjoint open neighborhoods. Examples include any compact topological group and any discrete topological group.
Totally disconnected topological groups: A topological group where the connected components are one-point sets. Examples include the p-adic integers (the completion of the rationals with respect to the p-adic metric) and any profinite group (the inverse limit of a system of finite groups).
Abelian topological groups: A topological group where the group operation is commutative. Examples include the real numbers with addition, the complex numbers on the unit circle with multiplication, and any finite Abelian group.
Non-Abelian topological groups: A topological group where the group operation is not commutative. Examples include the general linear group (the group of invertible linear transformations of a finite-dimensional vector space) and the quaternion group.
"The continuity condition for the group operations connects these two structures together."
"Consequently, they are not independent from each other."
"Topological groups have been studied extensively in the period of 1925 to 1940."
"Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups."
"Topological groups, along with continuous group actions, are used to study continuous symmetries."
"Continuous symmetries, which have many applications, for example, in physics."
"In functional analysis, every topological vector space is an additive topological group."
"With the additional property that scalar multiplication is continuous."
"Consequently, many results from the theory of topological groups can be applied to functional analysis."
"A combination of groups and topological spaces."
"The continuity condition for the group operations connects these two structures together."
"Haar and Weil showed that the integrals and Fourier series are special cases of a very wide class of topological groups."
"Topological groups have been studied extensively in the period of 1925 to 1940."
"To study continuous symmetries, which have many applications, for example, in physics."
"Scalar multiplication is continuous."
"Many results from the theory of topological groups can be applied to functional analysis."
"They are groups and topological spaces at the same time."
"They connect the concepts of groups and topological spaces together."
"They showed that integrals and Fourier series are special cases of a very wide class of topological groups."