TOPOLOGICAL GROUP Meaning and
Definition
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A topological group is a mathematical structure that combines the concepts of a group and a topological space. A group is a set along with a binary operation that satisfies certain properties such as closure, associativity, existence of identity element, and existence of inverse elements. A topological space, on the other hand, is a set equipped with a topology, which consists of a collection of open sets that satisfy certain axioms.
In a topological group, the group operation is also continuous with respect to the topology on the underlying space. This means that if we take two elements of the group and perform the group operation, the result will still be an element of the group, and this process can be continuously deformed without leaving the group.
The topological structure on a topological group allows us to define concepts such as continuity and convergence, which are not present in abstract algebra. By combining the group structure and the topological structure, we can study the interplay between algebraic and topological properties.
Some important examples of topological groups include the real numbers with addition, the circle group, and the general linear group. Topological groups find applications in various fields of mathematics, including algebraic topology, differential geometry, and harmonic analysis.
Overall, a topological group is a mathematical structure that combines the notions of algebraic group operations and topological continuity, allowing for the study of both algebraic and topological properties simultaneously.
Etymology of TOPOLOGICAL GROUP
The word "topological" is derived from the Greek word "topos" meaning "place" and the suffix "-logia" meaning "study" or "science". It refers to the branch of mathematics concerned with the properties of space that are preserved under continuous transformations, such as stretching or bending.
The word "group" in mathematics comes from the notion of a set with a binary operation that satisfies certain properties, such as closure, associativity, identity element, and inverse element.
Therefore, the term "topological group" combines the ideas of studying the properties of a space or place with respect to continuous transformations, together with the algebraic structure of a group. It refers to a group that is also equipped with a topology, where the group operations of multiplication and taking inverses are continuous functions with respect to that topology.
