Lie groups are mathematical structures that combine algebraic and geometric properties, allowing for the study of continuous symmetries. They are fundamental in many areas of mathematics and physics, providing a framework for analyzing objects that are invariant under transformations like rotations and translations.
Group actions provide a way to represent group elements as transformations of a set, allowing the study of symmetry in mathematical structures. They are fundamental in understanding the structure of groups and their influence on geometric and algebraic objects.
Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.