Introductory topology
Undergraduate · Math
Syllabus focus
Theoretical / proof-based
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$1,162 · Introductory topology · 18 tutoring hrs
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Topics typically covered
Theoretical / proof-based
Topological spaces
- Topological spaces: definition and examples (Euclidean, discrete, cofinite)
- Open and closed sets; interior, closure, and boundary
- Bases and subbases; subspace topology
- Continuous functions and homeomorphisms
- Product and quotient topologies (introduction)
Separation and countability
- Hausdorff spaces and separation axioms (T0–T4 overview)
- Limit points and derived sets
- First- and second-countability; separable spaces
- Metric spaces as topological spaces; equivalence of definitions
- Completeness in metric spaces (introduction)
Compactness and connectedness
- Compactness: open covers and sequential compactness in metric spaces
- Heine–Borel and Bolzano–Weierstrass in R^n
- Connected and path-connected spaces; components
- Compactness of continuous images and extreme value theorem (topological proof)
- Introduction to homotopy and the fundamental group (preview)
Notes
Topics reflect common introductory topology syllabi for mathematics majors at US universities. This course often follows real analysis and precedes graduate topology or algebraic topology.