Welcome!

Welcome to the Topology course website!

This course is an inquiry-based introduction to point-set and algebraic topology. You will investigate and prove all the results yourself, with guidance from a carefully curated set of notes and your instructor, and with collaboration with classmates.

What is topology? Roughly speaking, topology can be thought of as the study of "continuous deformations" of objects. Imagine an object that is made of a very stretchy rubber (or Play-Doh). A continuous deformation of the object is some warping (stretching, shrinking) of the object that doesn't involve tearing or gluing.

We will start by developing a sufficiently general notion of a continuous function using set theory. We will then consider the problem of identifying when two spaces are equivalent (i.e., when they can be continuously deformed into each other) using notions of compactness and connectedness. We will also learn about and explore the "separation axioms" that in some sense classify the degree to which topological spaces behave like Euclidean space. Finally, we will turn our attention to the set of paths (closed loops, specifically) in a topological space, and use this set of paths to construct what is known as the FUNdamental group of the space.

Course Learning Objectives

• Work with topological definitions and theorems related to the content described above.
• Read and evaluate the correctness of topological proofs.
• Produce examples and counterexamples that illustrate why theorem hypotheses are necessary or why a statement is untrue.
• Draw pictures to represent topological ideas.
• Formulate conjectures about topological concepts, and test these conjectures.
• Prove topological statements.
• Use topological ideas (e.g., homeomorphisms, fundamental group) to classify spaces.
• Present mathematical arguments both orally and in writing.