Theoretical / proof-based
Introductory topology · Undergraduate · Math
Learning objectives from the Introductory 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.
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)
Learning objectives
Click an objective for study materials.
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)
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.
- Topological spaces
Topological spaces
Coming soon - Separation and countability
Hausdorff spaces
Coming soon - Compactness and connectedness
Compactness
Coming soon
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$1,162 · Introductory topology · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.