Pacific Math Club Lecture Series: Topology

Ran a weekly lecture series through University of the Pacific’s mathematics club to lecture on topics from point-set topology which is not offered as a course at the University of the Pacific. Topics were presented in a way friendly to those with background in calculus at the levels of MATH 051 and MATH 053.

Resources used

Handouts written by myself that are inspired by or reference the following books:

• Real Mathematical Analysis by Pugh
• Introduction to Topological Manifolds by Lee

Topics Covered

1. Real analysis in \(\mathbb R\) and \(\mathbb R^n\)
   1. Sequences, subsequences, and convergence
   2. \(\varepsilon\)-\(\delta\) condition
   3. Continuity
2. Metric spaces
   1. Sequences, subsequences, and convergence
   2. \(\varepsilon\)-\(\delta\) condition
   3. Open subsets and closed subsets
   4. Continuity and homeomorpisms
   5. Metric subspaces and product spaces
3. Topological spaces
   1. Topology and open subsets
   2. Sequences, subsequences, and convergence
   3. Continuity and homeomorphisms
   4. Hausdorff Property
   5. Bases and countability properties
   6. Manifolds
   7. Subspaces
   8. Product spaces
   9. Quotient spaces
   10. Compact spaces
   11. Connected spaces