This subtopic studies topological algebraic systems, where algebraic operations are considered together with topological structure and continuity requirements.

This subtopic studies locally compact abelian groups, focusing on harmonic analysis, Pontryagin duality, and structural classification in the locally compact setting.

This subtopic studies compact groups, emphasizing structure theory, Haar measure, representation theory, and the role of compact symmetry.

This subtopic studies locally compact groups and their representations, including nonabelian harmonic analysis and the interaction between topology and unitary representation theory.

This subtopic studies Lie groups, connecting smooth manifold structure with group actions, differential geometry, and representation theory.

This subtopic studies transformation groups, where groups act on spaces and the resulting orbit, stabilizer, and symmetry structures are analyzed.