Math 655 - Introduction to Topology

Math. 655 is an introduction to the basic concepts of modern topology: metric spaces, topological spaces, connectedness, compactness, completeness, quotient spaces, manifolds, and classification of surfaces. While the course will emphasize the geometric aspects of topology, some applications to analysis will also be discussed, such as the Banach fixed point theorem and the existence of solutions to first order differential equations. Math. 655 is the first of a year long sequence. The followup courses Math. 656 and 657 will discuss fundamental groups and covering spaces and homology respectively.

Basic Course Related Information

Examples and Illustrations for Course Lectures

This section contains pictures and animations to supplement course lectures. Sometimes homework problems (and take-home final problems) will refer to some of these pictures.

  1. Early history of topology
  2. Euler's formula for polyhedra
  3. Picture for problem 5, Homework 1.
  4. Euler's formula in higher dimensions
  5. Pictures of various objects we will encounter in the study of topology.
  6. More pictures
  7. Space-filling curves
  8. The Möbius strip: an introduction to nonorientable surfaces
  9. The Klein bottle
  10. Visualizing the hypercube
  11. Crab hamburger or why nommanifolds are bad.
  12. Classification of closed surfaces
  13. Jordan curve theorem and its generalizations
  14. p-norm circles in the plane

For an alternative take on some of these topics, check out this innovative multi-disciplinary project by a group of Yale undergraduates: