Topology
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Topology — scope drawn from Bradley–Bryson–Terilla Topology (MIT Press open chapters) and typical graduate point-set / algebraic topology intro syllabi.
Point-set topology
- Topological spaces, bases, and subspaces — Topological spaces, bases, and subspaces
- Continuity and homeomorphisms — Continuity and homeomorphisms
- Product topologies — Product topologies
- Compactness: Heine–Borel, Tychonoff (introduction), sequential compactness — Compactness: Heine–Borel, Tychonoff (introduction), sequential compactness
- Connectedness and path connectedness — Connectedness and path connectedness
Separation and countability
- Hausdorff and regular spaces — Hausdorff and regular spaces
- Normal spaces (introduction) — Normal spaces (introduction)
- Urysohn lemma — Urysohn lemma
- Tietze extension (as covered) — Tietze extension (as covered)
- First and second countability — First and second countability
Algebraic topology introduction
- Homotopy and homotopy equivalence — Homotopy and homotopy equivalence
- Fundamental group — Fundamental group
- Van Kampen theorem (introduction) — Van Kampen theorem (introduction)
- Covering spaces (overview) — Covering spaces (overview)
Manifolds and geometry intro
- Topological and smooth manifolds (definitions and examples) — Topological and smooth manifolds (definitions and examples)
- Tangent spaces — Tangent spaces
- Vector fields (introduction) — Vector fields (introduction)
- Orientation and degree (overview) — Orientation and degree (overview)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Point-set topology
Topological spaces, bases, and subspaces
Coming soon - Separation and countability
Hausdorff and regular spaces
Coming soon - Algebraic topology introduction
Homotopy and homotopy equivalence
Coming soon - Manifolds and geometry intro
Topological and smooth manifolds (definitions and examples)
Coming soon
Notes
Graduate Topology — scope drawn from Bradley–Bryson–Terilla Topology (MIT Press open chapters) and typical graduate point-set / algebraic topology intro syllabi. Topic outline: `content/topics/graduate/topology.json`.