My professor had one of these stickers right on his office door. It’s a classic and simple topology joke, playing off the fact that topology is often regarded as the study of “nice” continuous deformations, such as stretching, shrinking, and twisting. The professor himself liked to describe topology as “the study of infinitely stretchy elastic bands.” And it only takes a few seconds on the Wiki page for topology to find a fascinating image of a mug into a doughnut.
So if you’re like me and you’ve seen and heard about all these cool things and think topology is the most hip form of geometry you’ve ever come across, then you might be surprised when you get to the definition of a topology:
Given a set X, a collection of subsets of X, denoted T, is a topology on X if:
- X and the empty set are in T
- Any arbitrary union of elements of T is also in T
- Any finite intersection of elements of T is also in T
WHAT. WHERE’S ALL THE COOL BENDY STRETCHY FLEXIBLE GEOMETRY WHAT IS THIS NONSENSE SET THEORY
Or at least that was my initial reaction… What you often see in a first undergraduate course, or sometimes even a graduate course, is known as point-set topology, and ends being a development of topology through a set-theoretic point of view, that dips in and out of geometry.
And this development, honestly, can be a bit disappointing. You probably weren’t looking to learn more about things you can do with sets, you probably don’t care that much about learning what an open set and closed set is, and you probably don’t want to learn about connectedness and compactness (which you may have already learned in your analysis class anyway). So it’s frustrating.
BUT DON’T GIVE UP. Depending on how fast your course moves, you may spend all semester on these set theoretic ideas. But as soon as you learn about continuity (and, honestly, even a little before if you’re willing to lose a little rigor), you could go straight on to learn about homotopies and the fundamental group and all sorts of cool algebraic topology things
And as you get to homeomorphisms, quotient maps, and quotient spaces (which immediately follow after continuity), you’re right there on the edge of all those “cool” things that you were expecting from the class.
So, don’t get too discouraged. Point-set topology might seem a bit boring and not quite what you were looking for at first, but if you’re ambitious and want to learn things on your own or can even push you professor into teaching some different things, then all the cool rubber geometry stuff is right around the corner, or at least awaiting you in your next class in topology. And remember, point-set topology may seem dull, but structures on sets are pretty fascinating, so enjoy it while you can!