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

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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


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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


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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend.

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Thank you for your support!


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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


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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...