MATH 5345H — Lecture Notes (Fall 2026)

Notes are posted as they are covered in class. Click a week to open the HTML version of the notes for that week.

• Week 1 (Sep 9) — Course overview and proof writing; sets, functions, relations, and the standard number systems (§§1–4)
• Week 2 (Sep 14, Sep 16) — Cartesian products; finite, countable, and uncountable sets (§§5–7); infinite sets, the axiom of choice, and well-ordered sets (§§9–10)
• Week 3 (Sep 21, Sep 23) — Topological spaces and standard examples (§12); bases, subbases, and comparison of topologies (§13)
• Week 4 (Sep 28, Sep 30) — The order topology and the finite product topology (§§14–15); subspaces, closed sets, closure, and limit points (§§16–17)
• Week 5 (Oct 5, Oct 7) — Continuity, homeomorphisms, and the pasting lemma (§18); arbitrary products and coordinate maps (§19)
• Week 6 (Oct 12) — Metric topologies, metrizability, and sequences (§§20–21)
• Week 7 (Oct 19, Oct 21) — The quotient topology and identification spaces (§22); connectedness and connected subspaces of the real line (§§23–24)
• Week 8 (Oct 26, Oct 28) — Components, path components, and local connectedness (§25); compactness and the finite intersection property (§26)
• Week 9 (Nov 2, Nov 4) — Compact subsets of Euclidean space; Heine–Borel and related results (§27); limit point compactness and sequential compactness (§28)
• Week 10 (Nov 9, Nov 11) — Local compactness and the one-point compactification (§29); countability and separation axioms; examples and counterexamples (§§30–31)
• Week 11 (Nov 16, Nov 18) — Normal spaces and the Urysohn lemma (§§32–33); the Urysohn metrization theorem and the Tietze extension theorem (§§34–35)
• Week 12 (Nov 23, Nov 25) — The Tychonoff theorem and compactness of products (§37); local finiteness and the Nagata–Smirnov metrization theorem (§§39–40)
• Week 13 (Nov 30, Dec 2) — Paracompactness and the Smirnov metrization theorem (§§41–42); complete metric spaces; compactness in metric spaces (§§43, 45)
• Week 14 (Dec 7, Dec 9) — Pointwise and compact convergence; the Ascoli theorem (§§46–47); Baire spaces and an introduction to dimension theory (§§48, 50)
• Week 15 (Dec 14) — Homotopy of paths; course review (§51)

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