Course Summary

One of the surprises in analysis is how useful notions like closed or open --- and especially compactness --- are in proving analytic statements. Indeed, after slogging through a few weeks of notions like these, one can typically move through the whole of single-variable differential calculus in about a week. The central place these concepts play in the development of such a fundamental mathematical discipline hints at how powerful these tools can be, and so it's natural to see where else these concepts might be useful. Math 307 (Topology) does precisely this, using the notion of openness as an axiomatic starting point for developing a theory that can sit entirely on its own. As with other abstractions performed in mathematics (e.g., eschewing $\mathbb{R}^n$ in favor of studying general $n$-dimensional vector spaces over $\mathbb{R}$), this axiomatic approach is both foreign and powerful: what does one lose when openness isn't defined in the way we're used to, and how do we both leverage and ignore our intuition in familiar topological settings?

We'll start with the central definition of the course --- topological space --- and develop a slew of familiar and unfamiliar examples. We'll see some concepts and behavior from analysis that hold in more general topological spaces, but we'll also see that topology has room for plenty of "pathological" examples that showcase just how nice $\mathbb{R}^n$ is (topologically speaking). We'll also see that there's a way to describe when two objects are "the same" topologically, and we'll examine the kinds of properties that are (and aren't!) preserved under this topological equivalence.

Course Instructor

The professor for this class is Andy Schultz. Office hours will be held on Zoom at the following times

The link for Zoom office hours is sent to the class directly via email at the start of the semester. You are highly encouraged to attend office hours, and you never need an appointment to do so. If these office hours don't fit with your schedule, contact the instructor so that he can either adjust when ``official" office hours are held or set up an appointment to help you outside of office hours.

You can contact the instructor at . Though he is always happy to receive emails from you with questions or concerns about the course, he can't guarantee that he'll be able to promptly reply to emails late at night or over the weekend. If you do contact the professor by email, please be sure to follow standard email etiquette. In particular, please make sure you include a greeting and signature and avoid abbreviations. If you're contacting him to ask about a problem, please be sure to specify what the problem asks (as opposed to asking something like ``I can't get problem 2 and need your help").