Introductory topology
Undergraduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Introductory Topology — scope drawn from open topology notes (metric/topological spaces) and typical US intro topology syllabi.
Topological spaces
- Topological spaces — Topological spaces
- Examples (Euclidean, discrete, cofinite) — Examples (Euclidean, discrete, cofinite)
- Open and closed sets — Open and closed sets; interior, closure, and boundary
- Bases and subbases — Bases and subbases; subspace topology
- Continuous functions and homeomorphisms — Continuous functions and homeomorphisms
- Product and quotient topologies (introduction) — Product and quotient topologies (introduction)
Separation and countability
- Hausdorff spaces — Hausdorff spaces
- Separation axioms (T0–T4 overview) — Separation axioms (T0–T4 overview)
- Limit points and derived sets — Limit points and derived sets
- First- and second-countability — First- and second-countability; separable spaces
- Metric spaces as topological spaces — Metric spaces as topological spaces; equivalence of definitions
- Completeness in metric spaces (introduction) — Completeness in metric spaces (introduction)
Compactness and connectedness
- Compactness — Compactness
- Sequential compactness in metric spaces — Sequential compactness in metric spaces
- Heine–Borel — Heine–Borel
- Bolzano–Weierstrass in R^n — Bolzano–Weierstrass in R^n
- Connected — Connected
- Path-connected spaces — Path-connected spaces
- Compactness of continuous images — Compactness of continuous images
- Extreme value theorem (topological proof) — Extreme value theorem (topological proof)
- Introduction to homotopy — Introduction to homotopy
- The fundamental group (preview) — The fundamental group (preview)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Topological spaces
Topological spaces
Coming soon - Separation and countability
Hausdorff spaces
Coming soon - Compactness and connectedness
Compactness
Coming soon
Notes
Undergraduate Introductory Topology — scope drawn from open topology notes (metric/topological spaces) and typical US intro topology syllabi. Topic outline: `content/topics/undergraduate/intro_topology.json`.