MATH 4530: Introduction to Topology (Fall 2023)

Topology is the study of topological spaces: point sets endowed with "nearness" information. These are extremly general objects which appear in many areas of mathematics. The purpose of this course is to develop our intuition about topological spaces as we study key examples, learn various constructions and prove fundamental theorems of topology.

This course, held Fall 2023 at Cornell University, will be divided into two parts: the first part will cover the basics of general point-set topology and the second part will introduce cellular complexes and algebraic topology.

Basic Information

• Instructor: Nima Hoda
  • Email: nima.hoda@cornell.edu
  • Office hours: Wednesdays 11:00am – 12:00pm and Thursdays 1:30pm – 2:30pm in 402 Malott Hall.
• TA: Anthony Graves-McCleary
  • Email: ag2537@cornell.edu
  • Office hours: Wednesdays and Fridays, 2:00pm – 3:00pm in 218 Malott Hall.
• Textbook: Topology (2nd Edition) by James R. Munkres
• Lectures: Tuesdays and Thursdays, 11:40am – 12:55pm in 112 Rockefeller Hall, Cornell University.

Resources

• Toplogy Hangout: Tuesdays, 4:25pm – 6:25pm in 206 Malott Hall.
  • Use this space to initiate group discussions on homework problems and lecture material.
• Ed Discussion: For online discussions of homework and lecture material.

Evaluation

• 20%: homework (posted on gradescope)
• 15%: prelim exam #1: in-class on Thu. Sep. 28
• 15%: prelim exam #2: in-class on Thu. Nov. 2
• 10%: end of semester presentation
• 40%: final exam

Schedule

* In-class preliminary exam on this date.

Presentation

Each student will prepare and give a 15 minute presentation on a topic of interest related to topology. Alongside their presentation, each student will also prepare a one-page handout on letter size (8.5 by 11 inch) paper summarizing their topic, to be made available to the class.

In order to avoid overlap, presentation topics are to be chosen in consultation with the instructor on a first come first served basis and should be fixed at the latest by Nov. 13th. Students are encouraged to choose their topic and begin work on their presentation well in advance of this date. A list of currently chosen topics can be found on the course Canvas.

Presentations will take place in meetings scheduled every weekday from 6:30pm – 8:30pm the week of Nov. 27 – Dec. 1 in 253 Malott Hall. Students who choose to present electronic slides must bring their own device equipped with an hdmi output port and are encouraged to test their device in 253 Malott Hall in advance of their presentation date. Students are required to attend the entire meeting in which their presentation is scheduled and are highly encouraged to attend all meetings. Any student attending at least half of the meetings will be awarded a bonus point on their presentation grade.

Presentation topic ideas

Below are some ideas for presentation topics.

Many of the following are from this page on Wikipedia. In general, Wikipedia is a great starting point for your presentation topic research.

• Explain an appropriate math blog post (e.g. from Dan Ma's Topology Blog or from the Wild Topology blog)
• Explain an interesting example from Counterexamples in Topology
• Explain an appropriate portion of a research or expository paper
• Inverse and direct limits
• Hausdorff dimension and fractals (e.g. Menger sponge, Sierpinski's triangle)
• The notion of a (smooth) atlas in defining manifolds and how this can be used to define additional structures on manifolds.
• Compactifications of spaces other than the one-point compactification (e.g. the Stone–Čech compactification)
• Nets as a generalization of sequences and how they do for general topological spaces what sequences do for metrizable spaces
• The Baire category theorem
• Paracompactness and sigma-compactness
• Urysohn Metrization Theorem and/or other metrization theorems
• Space-filling curves
• Polynomials and the Zariski topology
• The compact-open topology on a space of continuous maps
• Real trees and dendrites
• Ultraproducts and asymptotic cones of metric spaces
• Uniform spaces
• Stone's representation theorem for Boolean algebras
• Toplogical notions of dimension: inductive dimension, Lebesgue covering dimension
• The nerve complex and Borsuk's Nerve Theorem
• The Hawaiian earring
• The Seifert-Van Kampen Theorem
• The Hairy Ball Theorem
• The Tietze Extension Theorem
• Higher homotopy groups
• The Jordan Curve Theorem
• Planar graphs and Kuratowski's Theorem

Homework

Homework assignments will be posted to the MATH 4530 gradescope.