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The Concept So Much of Modern Math is Built On | Compactness

Go to https://brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription. Compactness is one of the most important concepts in Topology and Analysis, but it can feel a little mysterious and also contrived when you first learn about it. So what is compactness, intuitively? And why is it so fundamental to so much of modern math? =Chapters= 0:00 - Intro 2:26 - Formal Definition 3:03 - Topology Review 4:03 - Unpacking the Definition 6:33 - What Do Compact Sets Look Like? 8:02 - Sequential Compactness 10:03 - Making a Set Sequentially Compact 14:46 - What is Compactness Good For? 18:45 - Wrap Up 19:22 - Brilliant Ad =============================== The quote about compactness being a "gate-keeper" topic to math students comes from the paper "A pedagogical history of compactness" by Manya Raman-Sundstrom which provides, well, exactly what it says. You can find it here: https://arxiv.org/abs/1006.4131 =============================== This video was generously supported in part by these patrons on Patreon: Marshall Harrison, Michael OConnor, Mfriend. To support future videos, become a patron at   / morphocular   Thank you for your support! =============================== CREDITS The music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Rubix Cube, Falling Snow The track "Rubix Cube" comes courtesy of Audionautix.com =============================== The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here: https://github.com/morpho-matters/mor...