A topological group is a mathematical structure that combines the concepts of group theory and topology. It consists of a set equipped with a group operation that is continuous with respect to the topology on the set. This means that both the group operation and the inverse operation are continuous functions, allowing for a seamless interaction between algebraic and topological properties. Topological groups are important in various areas of mathematics, including algebraic topology and functional analysis. Examples include the real numbers under addition and the circle group, which consists of complex numbers of unit modulus under multiplication. These structures help mathematicians study symmetry and continuity in a unified framework.