Topology

Topology generalizes the notions of "closeness" and "continuity" you met in advanced calculus, stripping them down to the bare structure of open sets so that the same arguments work on spaces far weirder than R^n. The first half is point-set: you'll prove things about connectedness, compactness, and separation axioms, and learn to construct spaces with prescribed properties. The second half pivots to algebraic topology — simplicial complexes, homology, and the fundamental group — where you compute invariants that distinguish spaces up to deformation. Expect proof-heavy homeworks; the course is the gateway to differential topology, geometry, and anything involving manifolds.