Algebraic And Geometric Complements 2023/2024

Topologies and topological spaces. Topological subspaces. Euclidean topology. Open and closed sets. Interior, closure, accumulation points, derived sets, boundary, neighborhoods, dense subsets. Continuous functions. Limits and convergence. Homeomorphisms. Metric spaces. Normed spaces. Topology induced by the metric. Topology induced by the norm. Hausdorff topological spaces. Product of topological spaces. Product topology. Connected topological spaces. Path connected topological spaces. Compact topological spaces. Groups. Subgroups. Matrix groups. GL(n,R), SL(n,R), O(n,R), SO(n,R), GL(n,C), SL(n,C), U(n), SU(n). Topology of these groups. Relationship between orthogonal groups and isometries of Euclidean space. Symmetry groups. Tangent spaces. Lie algebras. The Lie algebras of the classical  Lie groups.