A topological group is a mathematical structure that combines the concepts of group theory and topology, where a set equipped with a group operation is also a topological space such that the group operations are continuous. This means that both the multiplication and inversion operations of the group are continuous functions with respect to the topology of the space, allowing for a rich interplay between algebraic and topological properties. Understanding topological groups is crucial when studying concepts like Haar measure and invariant integration, as these ideas often rely on the continuous nature of group actions and their structure.
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