An Introduction to Compact Sets

Compact sets are the foundation that modern mathematics is built on, and here we explore their definition and properties. Optimization problems are justified because continuous functions take maximum values on compact sets, our first spaces of functions get magnitudes and distance because of compact sets. And all of this is because Maurice Frechet wanted to generalize the work of Weierstrass.

//Books

//Exercises
- Prove the product to sum identities for exponentials first for rational numbers then for real numbers.

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0:00 Introduction
1:52 ChatGPT still can't math...
4:36 What is a compact set?
6:20 Compact sets are closed
8:32 Prisms are closed
9:20 The Heine Borel Theorem
10:17 Frechet's Definition
10:58 Wrap up

[skepley]

Compact sets are my go to example when I'm explaining to math students that math is all about getting the "right" definitions. Definitions can turn someone's PhD dissertation into an undergraduate homework exercise. As usual you somehow manage to explain it so well in such a short video. Great work man.

[imnimbusy]

Sets are way cooler than the maths in my school “sets” them out to be. Thanks for the video, I never knew there was a symbol for multiple unions!

[FeDeRiCo]

This video is great. Your reference to Hausdorff spaces was, in my opinion, brilliant. I had one of my mayor WTF moments when I learn that. I wish my teacher made more of those kind of comments, so one could learn which properties are "more fundamental".

[om]

your channel started to took off just around the time i started learning real analysis haha (lucky for me), awesome work!

[imrematajz]

❤with all that hand-waving your step counter must go through the roof while you are sitting still in a compact and bounded subset of your space.

[Cyclonus-fcxx]

It's a great day when you see this channel has uploaded, I see the algorithm has blessed this video more than the last hopefully it continues to do that for future ones this channel really deserves more attention

[hilalahmad]

Obviously your explanation about any topic is awesome.
Sir please give lecture series on Real Analysis and metric space.

[johnchristian]

Great video as always, thats a lot of information contained in a few minutes. The animations also were on point!

[incredulity]

Very good video, but the music doesn't fit at all😊

[TonyMontana-vprn]

I think that at 4:10 it refers to [0, 1) as a topological subspaces of the reals with the induced topology, ergo the whole space [0, 1) is a clopen set, so in particular it is closed. Is it plausible?

[michaelpicardmscphd]

Can you turn off the lousy music? How does that help with anything? A distraction.

[perappelgren]

"Like" isn't enough! ❤️

[KaliFissure]

Surface(cos(u/2)cos(v/2), cos(u/2)sin(v/2), sin(u)/2), u, 0, 2pi, v, 0, 4pi

The radially symmetric Klein bottle.
Notice the 4pi needed to complete the surface. This is half a sphere caught over its own phase inversion.
Being a single sided form once you so the outside then you do the inside. Like renormalization. Once you pass infinite you are on the other side but never changed direction exactly. 🖖

Time is a compactified dimension one single Planck second in size

Neutron decay cosmology. Inevitable.

[voroldrwarfff]

Hello I have a question : Im a math undergrad and I saw those analysis concepts in my analysis 1 course, so it seems to be introductory stuff. Can you tell me how the concept of compact sets is used by professional mathematicians like you in your work ?

[fanalysis]

Can you do some videos on real variable harmonic analysis? Something at the level of Stein's monograph on real variable methods, orthogonality, and oscillatory integrals?

[christophergame]

Americans pronounce French words with a final é sound as being accented on the final é sound. Even words such as Fréchet.

[HUEHUEUHEPony]

Please use gpt-4 and not bare chat-gpt