MATH 409 Topology

Overview

MATH 409 Topology is a disciplinary course for students in the Mathematics track of the Applied Mathematics and Computational Sciences major and an elective for students in other majors.

Course Description

MATH 409 is a first course in topology. Topology is a fundamental area of mathematics that studies continuity, connectivity, compactness, and shape. In addition to being an important field in its own right, topology is used throughout mathematics, in geometry, analysis, algebra, number theory, combinatorics, etc. Topology is also useful in many other scientific fields, like computer science, physics, and, as of late, machine learning. Studying topology develops abstract mathematical thinking and can benefit anyone looking to pursue advanced STEM studies.

This course is divided into several modules. The first module introduces topological concepts by studying metric spaces. The next module dives into abstract topological spaces: what they are, how they can be transformed, and different properties they can have. In the third module, we study the important concepts of compactness and connectedness in topological spaces, and how these relate to products and quotients of spaces. Finally, in the last module, we turn to algebraic topology by studying the fundamental group of a topological space and the elements of homotopy theory. It is this final module that relates to the well-known definition of topology as “stretchy geometry.”

Topics include topology of metric spaces, abstract topological spaces, open and closed sets, connectedness, compactness, continuity, completeness, subspaces, product and quotient spaces, separation axioms, homotopies of paths, the fundamental group, covering spaces, index theory, and applications (Borsuk-Ulam Theorem, Ham Sandwich Theorem, Fundamental Theorem of Algebra).

Syllabus

MATH 409 Syllabus (Spring 2025)

Literature

Introduction to Topology. Theodore W. Gamelin and Robert Everist Greene. Dover, 2nd edition (1999).