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.

- Original announcement of the course
- Course textbook: Basic Topology by M. A. Armstrong
- Course Syllabus
- Lecture Notes
- Homework Assignments
- Quiz Solutions
- Take-Home Final Problems

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.

- Early history of topology
- Euler's formula for polyhedra
- Picture for problem 5, Homework 1.
- Euler's formula in higher dimensions
- Pictures of various objects we will encounter in the study of topology.
- More pictures
- Space-filling curves
- The Möbius strip: an introduction to nonorientable surfaces
- The Klein bottle
- Visualizing the hypercube
- Crab hamburger or why nommanifolds are bad.
- Classification of closed surfaces
- Jordan curve theorem and its generalizations
- 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:

- Math That Makes You Go Wow
- local copy (with permission of authors)