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

Hey all. Just a few details I want to clarify: 1. Technically, the first version of the puzzle given at the beginning of the video (using an open domain) can be solved by a simple diagonal line since it attains its maximum and minimum at the domain's endpoints, which are excluded in this case. While that is true, I was trying to avoid that solution since it feels like one of those "technically correct" but not very interesting solutions since the graph is still bounded (yes, I could exclude this solution by using the term "supremum" instead of "maximum", but that's a term many viewers won't know and I didn't want to distract from the main point by explaining it). There's actually a footnote about this in the bottom-right corner at 0:54, but it flashed by quickly, so I don't blame you if you missed it. 2. Many have suggested using a constant function like f(x) = 3 as a solution to the second version of the puzzle (with a closed domain), but strictly speaking, a constant function DOES have both a maximum and a minimum: it's the value of the constant itself (e.g. 3). This is because the technical definition of "max" in math is the function's output that is greater than OR EQUAL TO all other output values, and likewise for "min". 3. At 10:24 I mentioned that a compact set can't have any "holes". This was meant to be casual speak for "single point gaps" and not large missing shapes in the interior. It's possible for sets to have "large holes" in them and still be compact. For example, as some have pointed out in the comments, an annulus (a disk with a smaller disk cut out of it) can be compact as long as it includes the entirety of both its inner and outer boundaries.

Hi. i am the one who wrote the paper cited here. Wow, I had no idea you had made a cartoon version of my master's thesis (more or less). Both humbled and amazed at how it has come to life. Wonderful job! ❤

I've been exposed to compactness in 3 different math classes, and I have to say this video has given me more clarity than anything else by an enormous margin

This is an unbelievably clear description of arguably one of the most abstruse concepts in mathematics. People who have not encountered compactness before are not going to appreciate just how special this video is. Absolutely stellar.

Back in university, a professor told me that compactness was an analysis/topology-friendly generalisation of finiteness. That has stuck with me since.

MOM NOT RIGHT NOW! I'M BUSY, NEW MORPHOCULAR VIDEO JUST DROPPED

Happy to see Topological concepts being explained so visually and intuitively :)

Glad you pointed out Seqential Compactness =/= Compactness in all topological spaces (metric spaces yes, others not necessarily). For those curious, there's an analogous idea to seqences called nets that *does* apply to all topological spaces: A topological space is compact iff every -bounded- net has a convergent subnet. Good intro to Compactness! It's definitely a weird one to wrap your head around initially, took me a good year of study before I was really comfortable with it and picking out when it'd be useful.

One of the big revelations for me when I was learning about covering spaces was that the bulk of the proofs were just compactness arguments. You're given a path in a space and some complicated local property for the points on that path. The proofs seem complicated, but they're just multiple iterations of "use compactness of the path to cut down open covers to finite subcovers where we can handle compatibility on overlapping sets".

The concept of compactness is one I've always known the definition(s) for, but had no real motivation to know or to have a thorough understanding of. This is a brilliantly compact and illustrative guide to it, and also a source of inspiration to look deeper into its further applications. Thank you for paying such meticulous attention to detail in making this, it has genuinely brightened my day and motivated me for the upcoming semester :>

After 3 years I have finally understood what "compact" means. Thank you

You have got to have one of the best math channels on RU-vid (or anywhere, for that matter). I know a lot of us (including myself) have been introduced to your channel via 3blue1brown's SoME challenges over the last couple of years, which puts you in some pretty stellar company, but this is truly top-tier stuff, man. This channel is equally as good and offers a much different approach as well as covers different topics at various levels of difficulty. Your ability to make such rigorous and counterintuitive concepts seem so natural is incredible. You are an awesome educator. Thank you for this.

Babe not now, a new morphocular video just dropped.

I had a question which I answered in the process of writing it down. I'm leaving it here in case somebody else is wondering the same thing. We have the [0, 1] interval and an infinite open cover consisting of balls where each consecutive ball gets smaller. More precisely the first covers [0, 0.5]; the second [0.5, 0.75]; the third [0.75, 0.875], etc. The first ball would have radius 0.25 + ε; the second 0.125 + ε/2; etc. What is the finite subcover? answer: This is not a cover because the point at 1 is not covered. If you try to cover 1, the new ball would cover some points near 1 and the remaining part of the line segment can be covered with finitely many balls.

Compactness always kind of baffeled me. In the finite-dimensional real numbers, it simply means 'closed and bounded', which I could wrap my head around. But in other domains, it means a little something extra that I could never really grasp. I am through with analysis already, but it was super nice to revisit this and get a better understanding for it!

At 3:57, I just want to add one comment. Open and closed are not opposites in topology. It is possible for a set to have no boundary points and be both open and closed.

wish i could send this video to myself 25 years ago! seriously, thank you for the intuitive explanation with excellent visuals

I think it’s fun that you can strengthen the boundedness claim to the full extreme value theorem straightforwardly with a second application of compactness, by showing that continuous images of compact sets are compact, and thus in particular closed (so the function’s range contains the minimum and maximum values) 🙂

Such a sophisticated domain and so lightly exposed with such thoroughness. Never had the chance to meet the beast before, just heard of it. Not so frightening after all 😅😉

Thank you for this excellent video ! I'm a self-learner and compactness was NEVER explained intuitively so with time I just accepted it to be the generalization of "close and bounded" (like open sets with open balls) without any motivation to the definition. Now, with this new point of view, the sequential approach make me understand a bit more the WHY

I cannot understate the value of what you are doing. I’ve always thought my brain was wired but math, but have struggled translating the concepts into language and vice versa; you do this masterfully. Thank you and subbed!

My classmates always wonder why I seem to get concepts so quickly in class, little do they know that I essentially watch hours of extra math lectures a week

Great content, as usual. Please keep making such valuable videos, the quality of your work is amazing.

I love how you start out with an example of how it is useful

what a fantastic video! i've had topology in class last year and i still learned something new! will be recommending this to my friends. keep up the great work! it's really appreciated

The proof of the Heine-Borel theorem is actually really nice, if you compare it to how transfinite induction works, you'll notice that it's basically like a topological version of induction for the real numbers. Edit: I'll expand on this: In the natural numbers, induction can be described by this idea: suppose you have a set S ⊂ ℕ, and it satisfies the following two conditions: - S contains 0 - if S contains all numbers lower than n, then S contains n. Then it follows that S must be the entirety of ℕ. Obviously this wouldn't directly work on the real numbers, but you can modify, it. Let's say you have a set S ⊂ [0, ∞), where [0, ∞) are all the nonnegative real numbers, and it satisfies the following two properties: - S contains an open set containing 0 - if S contains all numbers lower than n ∈ ℝ, then S contains an open set containing n Then it follows that S must be the entirety of [0, ∞). This can be verified to actually work through a simple least upper bound argument. If you replace [0, ∞) by a set of the form [0, x], you get a similar result that x must be contained in S. Now let's say you have an open cover U of the set [0, 1], and let S be the set of all numbers x such that [0, x] can be covered by a finite amount of sets from U. Then, - Clearly S contains an open set containing 0, as the cover must have an open set containing 0 and that single set forms a finite subcover of [0, e] for any sufficiently small e - if S contains all numbers lower than n, then find any open set A from the cover which contains n, by definition of open sets, you can find a number e such that (n-e, n+e) is contained in A, then by the assumption, we can find a finite subcover of [0, n-e] and by adding A to that finite subcover, we get a finite subcover of [0, n+e/2], therefore, an open set containing n, (n-e, n+e/2) is in S. Since we fulfilled the two conditions, we get that S contains 1, and therefore [0, 1] has a finite subcover.

I was rewarded by rewatching some segments of this video before moving onto the next ones. Very cool to learn how a topic I have been introduced to, analysis in one variable, is a special case of something much vaster.

Its amazing how we can have these quality of videos explaning the most complex ideas and we can access free in any place at any time. Thanks a lot

This is an excellent video, and just in time for my Analysis II exam! In Analysis I, I was introduced to compactness (together with the extreme value theorem and some other important concepts) only through sequential compactness in the context of metric spaces, and in Analysis II was stumped with the general definition of compactness, together with a (horrendously large) proof that it is equivalent to sequential compactness and also completeness and total boundedness on metric spaces. It's really hard to get an intuition for what it really does, I got a sense of the same kind of "reduction to finiteness" meaning when i spend hours picking apart the equivalence proof, but still until now compactness (at least the "finite subcover" version) was just some (pretty hard to understand) concept floating around in my head, and in not even 20 minutes you have given it a really good general meaning to keep with it! You've earned a new subscriber.

Amazing introduction to compactness. I consider myself already quite familiar with the concept, but this made it even more tangible and visual

Thanks for the clear exposition!

This is the most satisfying explanation of compactness I have ever gotten, brilliantly done!

This was pretty much my first introduction to topology so I definitely needed to pause the video and do some thinking for myself to wrap my head around some of these topics, but I thought this was such a great video!

This video is so good, I appreciate the efforts in both the explanation and the visuals The time spent to make this must have been crazy

This is possibly the best math video on RU-vid. Well done!

The example and the counter example of compactness in terms of open cover in the intervals [0, 1] and (0, 1) help me a lot in grasping the definition of compact space in terms of open cover. I understand both rigorously and intuitively about the sequential compactness since long ago, but I couldn't get the intuitive sense of open cover compactness even though I know its formal definition, right until I watched your presentation in the example in [0, 1] and the counter example in (0, 1). Thanks a lot! Keep up your brilliant work, Sir!

It's been many years since I took Analysis but this is the first time I've understood the motivation behind compactness.

i knew nothing about compactness, topology and next to nothing about analysis. Now I understand. Thank you.

this is so well done. i'm not sure if i know any other words that can describe how good this video is

congrats on nearly 100k, fully deserved

COMPACTNESS looks like a cool new SNES title the way you formatted the concept. Very clearly explained as always, Morph!

Literally studying topology rn. This is perfect timing 🎉🎉

Wow oh wow! When I first learned about Taylor and Maclaurin series (some 50 years ago) I was exposed merely to the mechanics of the functions and theorems. This actually gives insight to the behaviour of the functions on various domains. Thank you very much.

Wow that came full circle, also I got flash-backs to my intro to differential calculus class haha. Great video mate!

Really well made video. Props

I never really understood this concept when I took Real Analysis. Thanks for making it clear to me!

I have my bachelor’s and master’s in math, though I spent the last year studying statistics, and not analysis, so I am officially subscribing as this was a good review of what I spent years learning (and then a year forgetting in exchange for skills in quantitative methods)

Is it true that in reality mathematical concepts go inverse way from explanation - first a mathematician glances an intuition of what "that" is, and then refines and refines it to the point of definition, so "grasping it" (however vague) comes first, and definition actually comes last?

Brilliant explanation of one of Real Analysis' most challenging concepts. Well done!

Keep the topology videos coming! They are awesome

this is actually making me start to understand some stuff we were just supposed to assume in my intro linear algebra

Great work on this, thank you.

I honestly think this is the best new math youtube channel. Its like 3b1b yet has its own style and isn’t like those hundreds of 3b1b knock off channels that exist

Just learned Bolzano Weierstrass in our Real Analysis class, this is really helpful! Thanks!

Loved the video! I've struggled with compactness in real analysis and just automatically translated it to "closed and bounded". Animations and topology are a perfect match.

Hadn't even heard of compactness before. Thank you for the video.

More videos like this, please! To illustrate important math concepts.

I love your videos! Thanks for that!

This is a masterpiece! Thank you!

Beautifully explained

awesome visualization!!!

This was straight up better than my Real Analysis and Topology classes

One of the best math channels in the galaxy

3:58 in line with the terminology for doors, I propose a set that contains some of its boundary is called “ajar”

Congratulations on reaching 100K subscribers!

So great. Thank you. 🎉

Excellent and intuitive explanation :)

Thank you so much for making this. Love you😘😘😘😘😘😘

Best video I've ever seen in my entire life. I love you

This is a perfect video and it definitely achieves its purpose

Amazing video, thanks!

Can't wait to have a deeper grasp on compactness.

So much clearer than my analysis teacher!

Your way of explaining mathematics is compact , literally

Excellent explanation! 😊🎉

EVT proof at the end was great, thanks.

One of the most beautiful and useful concept in analysis

I was quite happy to see the technicality remark in the beginning, I was about to sperg out on stuff you'd most likely talk about later in the vid. Yes I like topology(not the knot subset, more the 'yes it is compact therefore the proof is done') Edit: there are ways of proving that compactness works differently In infinite dimensions spaces that don't require the choice axiom, and I've been thaught to use it as little as possible since it can lead to weird results. I recognise that using a base makes the proof simpler than sequences of continuous functions, but I prefer when people say they're bringing the big guns.

Oh my gosh! This is such a good vid! Way better job describing compactness than my advanced calc teacher lol

Brings back the memories!

A set that contains only part of its boundary is called "ajar" At least, that's what I called it when I took point set topoology in college.;-)

Man this is beautiful. Thanks.

The transition at 11:21 is just so smooth😊

Wonderful video. I loved it.

For humanity's sake, you need to make more videos like this one. Great job! I applaud you,

Damn man, that was a brilliant ad

Great video as always :)

Extreme Value this channel is

thanks, that really helps with the exam I have in two months

it is really frustrating in a way that I was able to pass classes like real analysis without ever really grasping the concepts intuitively

From my experience, I would rank compactness as the most important concept in all of mathematics. Other ones close to it would be curvature, convexity, and exact sequences. Background: phd in functional analysis and pdes

Ohh Jesus Christ my favourite channel's video just premiered..close all the doors, switch off all the lights..put on the headphone...ohh yes don't forget all other thoughts in your mind...take a deep breadth..enjoy the ride like no other!! Yes..yes..yes...

This compactness has some relation to Noetherian topological spaces like Noetherian schemes and other objects like presheaves, sheaves, coherent sheaves. Also of note is that locally compact topological groups have the averaging Haar measure and are used in harmonic analysis.

I remember a tiny, dense tome by Michael Spivac, I believe it was titled, Calculus on Manifolds, whose first chapter dealt with compactness, in all the frugal clarity that the printed page offers to illuminating math concepts. Needless to say, I did not get past page 2. That was back in 1988. Now, a return of Compactness via this video, and a part of my soul can now rest.

Excellent video 👏

This video explained to me why ford circles are important for N, and I am quite happy about that.

Man, really cool explanation

Brilliant video!

Great video! But just pointing a few things: 1. Talking about dimensions (in the usual sense) in metric spaces does not make sense, since it is a concept which you loose when you jump from normed spaces to metric spaces (noticing you used a norm instead of a metric when you showed the sequence in an infinite dimension space) 2. In metric spaces (even R^n with a given metric), closed and bounded is not enough for compactness, I can show a few examples if you want. Heine-Borell theorem only works with Euclidean (or equivalent) metrics. 3. There are topological spaces where you can fin compact sets that are not closed. The correct way to say it is: For a topological T2 space, compact implies closed. In metric spaces compact always implies closed since every metric space is T2. Sorry if my English isnt too good. Nice video!!

I appreciate it a lot!