What's the Difference Between Space and Set?

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If i wanted to define a topological space formally in mathematics, i would start with a set of points. So that could be a function that tells you how to compute the distance between two points. That's a metric space, which is an important type of topological space. Or there's a more abstract way to get at the same thing, in the absence of a formula for distances, ok? And so a line is a space. A circle is a space, a sphere. You can have mobi spans. An a mobis strip. Whether you're on the inside or the outside, there's no way to decide that. Wereson, you know, the toilet paper

“A way that math can make the world a better place is by making it a more interesting place to be a conscious being.” So says mathematician Emily Riehl near the start of this episode, and it’s a good summary of what’s to come. Emily works in realms of topology and category theory that are far away from practical applications, or even to some non-practical areas of theoretical physics. But they help us think about what is possible and how everything fits together, and what’s more interesting than that? We talk about what topology is, the specific example of homotopy — how things deform into other things — and how thinking about that leads us into groups, rings, groupoids, and ultimately to category theory, the most abstract of them all.

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Emily Riehl received a Ph.D in mathematics from the University of Chicago. She is currently an associate professor of mathematics at Johns Hopkins University. Among her honors are the JHU President’s Frontier Award and the Joan & Joseph Birman Research Prize. She is author of Categorical Homotopy Theory, and co-author of the upcoming Elements of ∞-Category Theory. She competed on the United States women’s national Australian rules football team, where she served as vice-captain.

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