Topological groups

Topological groups are a mathematical structure that combines concepts from both topology and group theory. Specifically, a topological group is a set equipped with two structures: a group structure and a topology, such that the group operations (multiplication and taking inverses) are continuous with respect to the topology.

Discrete groups are a type of mathematical structure studied primarily in the fields of abstract algebra and topology. Here's a breakdown of the concept: ### Definition A **discrete group** is a group \( G \) that is equipped with a discrete topology. In simpler terms, the group is a set of elements along with a binary operation (e.g.

An **amenable group** is a type of mathematical structure studied in the field of group theory, specifically in the study of topological groups and functional analysis. The concept of amenability is related to the ability of a group to have a certain type of "invariance" property under averaging processes. A group \( G \) is called **amenable** if it has a left-invariant mean.

Bohr compactification is a mathematical construction in the field of topological groups, particularly in the area of harmonic analysis and the theory of locally compact abelian groups. It is primarily associated with the study of the structure of such groups and their representations.

The Cantor cube, often denoted as \(2^\omega\) or \([0, 1]^\omega\), is a product space that arises in topology and set theory. It can be understood in a few different ways: 1. **Composition**: The Cantor cube is defined as the countable infinite product of the discrete space \(\{0, 1\}\).

Chabauty topology is a concept used in algebraic geometry and arithmetic geometry, specifically in the study of the spaces of subvarieties of algebraic varieties. It is named after the mathematician Claude Chabauty, who developed this topology in the context of algebraic varieties and their rational points. In the Chabauty topology, one can think about the space of closed subsets of a given topological space (often within a certain context such as algebraic varieties).

In the context of mathematics, specifically in the field of topology and group theory, a **compact group** is a group that is both compact as a topological space and a group in the sense of group operations. ### Definitions 1. **Topological Group**: A topological group is a set equipped with a group structure that is also a topological space, such that the group operations (multiplication and taking inverses) are continuous with respect to the topology of the space.

A **compactly generated group** is a type of topological group that can be characterized by the manner in which it is generated by compact subsets. Specifically, a topological group \( G \) is said to be compactly generated if there exists a compact subset \( K \subseteq G \) such that the whole group \( G \) can be expressed as the closure of the subgroup generated by \( K \).

A **continuous group action** is a mathematical concept that arises in the field of topology and group theory. Specifically, it involves a group acting on a topological space in a way that is compatible with the topological structure of that space. ### Definition: Let \( G \) be a topological group and \( X \) be a topological space.

In group theory, a branch of abstract algebra, a **covering group** is a concept that relates to the idea of covering spaces in topology, though it is used more specifically in the context of group representations and algebraic structures. A covering group can refer to a group that serves as a double cover of another group in the sense of group homomorphisms.

In the context of topological groups, the **direct sum** (often referred to as the **direct product**, especially in the category of groups) of a family of topological groups provides a way to combine these groups into a new topological group. The construction is analogous to that of the direct sum in vector spaces.

In the context of group theory, a *discontinuous group* usually refers to a group of transformations that is not continuous in a topological sense. This term can have different meanings depending on the mathematical context in which it is used, but here are two key interpretations: 1. **Mathematical Groups and Topology**: In general topology, a discontinuous group may refer to a group of homeomorphisms that do not form a continuous path between their elements.

In the context of topology and abstract algebra, an **extension** of a topological group refers to a way of constructing a new topological group from a known one by incorporating additional structure. This often involves creating a new group whose structure represents a combination of an existing group and a simpler group.

The Haar measure is an important concept in the area of harmonic analysis and abstract algebra, specifically in the context of topological groups. It is a way of defining a measure on a locally compact topological group that is left-invariant (or right-invariant), which means it remains unchanged (invariant) under the group's operations.

The Hilbert–Smith conjecture is a statement in the field of topology, particularly concerning group actions on topological spaces.

A homogeneous space is a mathematical structure that exhibits a high degree of symmetry. More formally, in the context of geometry and algebra, a homogeneous space can be defined as follows: 1. **Definition**: A space \(X\) is called a homogeneous space if for any two points \(x, y \in X\), there exists a symmetry operation (usually described by a group action) that maps \(x\) to \(y\).

The term "Identity component" can refer to different concepts depending on the context in which it is used. Here are a few interpretations across various fields: 1. **Mathematics**: In topology and algebra, the identity component of a topological space is the maximal connected subspace that contains the identity element. For a Lie group or a topological group, the identity component is the set of elements that can be path-connected to the identity element of the group.

Kazhdan's property (T) is a property of groups that was introduced by the mathematician David Kazhdan in the context of representation theory and geometric group theory. It is a strong form of compactness that relates to the representation theory of groups, particularly in how they act on Hilbert spaces.

Kronecker's theorem, also known as the Kronecker limit formula, is a result in number theory specifically related to the distribution of prime numbers and the behavior of certain algebraic objects. It can be particularly focused on the context of the theory of partitions or modular forms, but the term might refer to different results depending on the field.

A **locally compact group** is a type of topological group that has the property of local compactness in addition to the group structure. Let's break down the definitions: 1. **Topological Group**: A group \( G \) is equipped with a topology such that both the group operation (multiplication) and the inverse operation are continuous.

A **locally profinite group** is a type of group that is constructed from profinite groups, which are groups that are isomorphic to an inverse limit of finite groups. Formally, a locally profinite group can be defined as a group \( G \) that has a neighborhood basis at the identity consisting of open subgroups that are profinite.

A **loop group** is a concept from mathematics, particularly in the fields of algebraic geometry, differential geometry, and mathematical physics. It typically refers to a specific kind of group associated with loops in a manifold, particularly in the context of Lie groups.

In the context of Lie groups and algebraic groups, a **maximal compact subgroup** is a specific type of subgroup that has particular significance in the study of group structures. ### Definition: A **maximal compact subgroup** of a Lie group \( G \) is a compact subgroup \( K \) of \( G \) such that there is no other compact subgroup \( H \) of \( G \) that properly contains \( K \) (i.e.

A **monothetic group** is a term used in the context of taxonomy and systematics, particularly in the classification of organisms. It refers to a group of organisms that are united by a single common characteristic or a single attribute that defines that group. This characteristic is often a specific trait or combination of traits that all members of the group share, distinguishing them from organisms outside the group.

A **one-parameter group** is a mathematical concept primarily used in the fields of group theory and differential equations. It represents a continuous group of transformations that can be parametrized by a single real parameter, often denoted as \( t \).

A **paratopological group** is a mathematical structure that combines the concepts of group theory and topology, but with a relaxed condition on the topology. Specifically, a paratopological group is a set equipped with a group operation that is continuous in a weaker sense than standard topological groups.

The Peter–Weyl theorem is a fundamental result in the representation theory of compact topological groups. It describes how the regular representation of a compact group can be decomposed into irreducible representations. Here's a brief overview of the main points of the theorem: 1. **Compact Groups**: The theorem applies specifically to compact groups, which are groups that are also compact topological spaces. Examples include \(SU(n)\), \(SO(n)\), and \(U(n)\).

Positive real numbers are the set of numbers that are greater than zero and belong to the set of real numbers. This includes all the numbers on the number line to the right of zero, which can be represented as: - All whole numbers greater than zero (1, 2, 3, ...) - All fractions greater than zero (such as 1/2, 3/4, etc.) - All decimal numbers greater than zero (like 0.1, 2.

Quasiregular representation is a concept from the field of geometry and complex analysis, specifically within the study of quasiregular mappings. Quasiregular mappings are a generalization of holomorphic (complex analytic) functions, which allow for a broader class of functions including those that are not necessarily differentiable in the classical sense.

A "restricted product" typically refers to items that are subject to certain legal or regulatory limitations regarding their sale, distribution, or use. The specifics can vary widely depending on the context and jurisdiction, but here are some common categories of restricted products: 1. **Controlled Substances**: Pharmaceuticals or chemicals that are regulated due to their potential for abuse or harm (e.g., narcotics).

The Schwartz–Bruhat function, often simply referred to as the Schwartz function, is a type of smooth function that is rapidly decreasing. Specifically, it belongs to the space of smooth functions that decay faster than any polynomial as one approaches infinity. This type of function is especially important in various areas of analysis, particularly in the fields of distribution theory, Fourier analysis, and partial differential equations.

A **semitopological group** is a type of mathematical structure that combines aspects of group theory and topology. Specifically, it is a group \( G \) that is equipped with a topology such that the group operations, namely multiplication and taking inverses, satisfy specific continuity conditions, but not all of the usual requirements of a topological group.

A **topological group** is a mathematical structure that combines the concepts of a group and a topological space. Specifically, a topological group is a set equipped with two structures: a group structure and a topology that makes the group operations continuous.

A **topological ring** is a mathematical structure that combines the concepts of a ring and a topology. Specifically, a topological ring is a ring \( R \) that is also equipped with a topology such that the ring operations (addition and multiplication) are continuous with respect to that topology.

A totally disconnected group is a type of topological group in which the only connected subsets are the singletons, meaning that the only connected subsets of the group consist of individual points. This concept can be understood in the context of topological spaces and group theory. In more formal terms, a topological group \( G \) is said to be totally disconnected if for every two distinct points in \( G \), there exists a neighborhood around each point such that these neighborhoods do not intersect.