Topology

Standard topology topics from the canonical curriculum.

Core Topics

• Topological Spaces & Open Sets — The three axioms a collection of open sets must satisfy.
• Basis for a Topology — Generate a topology from a basis and compare two topologies.
• Subspace, Product & Quotient Topologies — Three canonical ways to build new spaces from old ones.
• Closure, Interior & Limit Points — Closure adds touched points, interior keeps surrounded ones, limit points are approached.
• Continuity in Topological Spaces — Continuous means preimages of open sets are open.
• Homeomorphisms & Topological Equivalence — When two spaces count as the same, and the invariants that tell them apart.
• Metric Topology — Open balls as a basis, and when a set counts as open.
• Connectedness — Separations, clopen sets, and the topological route to the Intermediate Value Theorem.
• Compactness — Open covers, finite subcovers, Heine–Borel, and the extreme value theorem.
• Separation Axioms (Hausdorff & beyond) — The T_0 through T_4 hierarchy: points, closed sets, and disjoint open separation.
• Homotopy of Maps & Paths — Continuous deformation of maps, path homotopy rel endpoints, and the equivalence relation.
• The Fundamental Group — Loops up to homotopy: the first invariant that detects holes.
• Covering Spaces — Path lifting, the degree, and computing the fundamental group of the circle.
• Euler Characteristic & Surface Classification — Genus, orientability, and = V - E + F classifying closed surfaces.
• Hairy Ball Theorem — You can't comb a hairy ball flat — a tangent field on S^2 must vanish.

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