Math 3200 – Introduction to Topology
In Spring semester 2016, I taught Math 3200: Introduction to Topology.
Office hours:
Tuesdays 1:00–2:30pm Wednesdays 1:00–2:00pm Thursdays 11:30–12:00pm and by appointment
Information:
Term: Spring 2016 Time: TR 2:35–3:50pm Location: 1120 Stevenson Center
Links:
View the Course Syllabus for the course description and course policies. View the Assignments Page for the list of homework assignments and some Challenge Problems. Access our Blackboard page (login required) to find your Grades for the class and other course materials.
Announcements:
- There was a small typo in Homework 13: In Problem 9, the relation should read \(t^{-1} V_{L+} - t V_{L-} + (t^{-1/2} - t^{1/2})V_{L_0}=0\) (the original version was missing the "\(=0\)" at the end). This has been corrected here and on blackboard.
- The Challenge Problems have been posted to the Assignments Page.
- Effective Monday, Feb 1, my office hours have been changed to Tues 1:00–2:30, Wed 1:00–2:00, and Thurs 11:30–12:00. Please plan accordingly.
Schedule:
Here is a tentative and ever-evolving schedule of the topics covered in each class. I will aim to have the topics for each class posted at least a class period in advance, and will modify the schedule after the fact to reflect what was actually covered.
Tues Jan 12 — Introduction; Distribute and discuss syllabus; Basic set theory and functions; Euclidean spaces Thur Jan 14 — Euclidean spaces; Metric spaces Tues Jan 19 — Open and closed sets in metric spaces; Binary space Thur Jan 21 — Definition of a topological space; Examples of topological spaces; Basic properties Tues Jan 26 — Basis; Topology generated by a basis; More examples of topologies; Interior and Closure Thur Jan 28 — Interior and Closure; Examples of these in various topologies; Limit points Tues Feb 2 — Limit points; Limit Point Theorem; Closure Theorem; Sequences; Definition of convergence Thur Feb 4 — Sequence Propositions 1–3; Hausdorff spaces; Example of sequences with multiple limits Tues Feb 9 — Functions; Injectivity/Surjectivity; Cardinality; Countable and Uncountable sets Thur Feb 11 — Cantor's diagonalization argument; Continuity; Characterizing continuity Tues Feb 16 — Midterm Exam Thur Feb 18 — Continuity: Examples and basic properties; Continuity in metric spaces Tues Feb 23 — Continuity and metric spaces; subspace topology Thur Feb 25 — Subspace topology; Product topology; Examples Tues Mar 1 — Return and discuss exams; Product Topology and examples; Connectedness Thur Mar 3 — Connectedness and subsets, products, & continuity; Connectedness of the real numbers Tues Mar 15 — Intermediate Value Theorem and applications; Cut points; Path connectedness Thur Mar 17 — Open covers; Compactness and examples; Continuous image of compact space is compact Tues Mar 22 — Compact vs Closed (+Hausdorff); Products and compactness; Nested Interval Lemma Thur Mar 24 — Closed intervals are compact; Heine-Borel Theorem; Extreme Value Theorem; Subsequences Tues Mar 29 — Compactness and convergent subsequences; the middle-third Cantor Set Thur Mar 31 — The Cantor Set, Binary Space, and a space-filling curve! Tues April 5 — Knot theory; Embeddings; Homotopy, isotopy, ambient isotopy; Knot equivalence Thur April 7 — Knot projections, shadows, and diagrams; Reidemeister moves; Links; Writhe; Linking number Tues April 12 — Knot invariants; 3-colorability; Crossing and Unknotting numbers; Knot polynomials Thur April 14 — Bracket polynomial: Definition; examples; invariance under Reidemeister moves R0, R2, R3