Theoretical / proof-based
Topology · Graduate · Math
Learning objectives from the Topology syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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 compac... — 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)
Definitions and structure
- 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 compac... — Compactness: Heine–Borel, Tychonoff (introduction), sequential compactness
- Connectedness and path connectedness — Connectedness and path connectedness
Proofs and reasoning
- 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
Abstraction and generalization
- 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)
- Topological and smooth manifolds (definitions and examples) — Topological and smooth manifolds (definitions and examples)
Modeling and computation
- Apply topological spaces, bases, and subspaces in engineering contexts — Apply topological spaces, bases, and subspaces in engineering contexts
- Apply continuity and homeomorphisms in engineering contexts — Apply continuity and homeomorphisms in engineering contexts
- Apply product topologies in engineering contexts — Apply product topologies in engineering contexts
- Apply compactness: heine–borel, tychonoff (introduction), sequential... — Apply compactness: heine–borel, tychonoff (introduction), sequential compactness in engineering contexts
- Apply connectedness and path connectedness in engineering contexts — Apply connectedness and path connectedness in engineering contexts
Data and technology
- Use software to explore topology problems numerically — Use software to explore topology problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- 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
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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