On the surface of a doughnut, there are fundamentally two different ways to wrap a closed path on the surface (through the whole or around the ring); every possible way is a combination of these two. This doesn’t depend on the exact dimensions of the doughnut. For another example, a loop of string tied into a trefoil knot can’t be deformed into a square knot. Topology can be considered the study of properties of shape that don’t change under changes of relative size or deformation. The tricky thing is to make precise mathematics out of this. There are surprising results: the wind velocity on the surface of a spherical planet must be exactly still in at least two places. If you take a piece of paper out of your notebook, crumple it, and place it back on the notebook, at least one point on the crumpled paper must be exactly over the position it started. There are 24 ways to stretch an 8-dimensional sphere over a 5-dimensional sphere, but only 2 ways over a 6-dimensional one. These abstract results often have powerful practical consequences. Topology can also be considered the abstraction of the idea of limits and convergence in calculus and analysis.

The first half of this class will rigorously develop the subject of topology (using metric spaces) from the axioms. The development will be abstract and proof-based. In this sense, the class is a good sequel to *Logic, Proofs, Algebra, and Set Theory*. The second half will survey how these concepts can be applied to geometric examples like the ones mentioned above, sometimes omitting proofs as needed; in that respect, the class is a good sequel to *Geometry*.

Course Level: 4000-level

Credits: 4

M/Th 1:40PM-5:20PM (2nd seven weeks)

Maximum Enrollment: 20

Course Frequency: Every 2-3 years

Categories: All courses , Mathematics

Tags: computer programming