The Concept So Much of Modern Math is Built On | Compactness
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- čas přidán 31. 05. 2024
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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:
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.
To support future videos, become a patron at / morphocular
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:
github.com/morpho-matters/mor...
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.
10:35 if a sequence approaches a single gap-point, doesn't is converge to this point even if the point is not in the set?
@@LB-qr7nv Ah, I should have said "fail to converge *to a point in the set*". Yes, if the sequence approaches a single point gap, it will still converge to that point (assuming you're including that gap point as part of the larger ambient space), but the set won't be sequentially compact since the limit point is outside the set.
I was thinking more the absolute value of a tangent.
@@user-te3ii8ru1m It actually can't be solved using a straight line. The extreme value theorem tells us that on a closed interval [a,b] there must be one maximum and one minimum. It is never stated that one point can't be both a min and a max at the same time, as the theorem only fails when there is either a higher or a lower point than our original point. If you find that a bit hard to believe, then imagine a function f(x) = 2. This function is always constant, right? You can ask yourself the following "What is the lowest possible value for f?" you can instantly realize that "Well, I can't go lower than 2, therefore it is the lowest possible value". The same reasoning works for the maximum value.
It's been a while since I learned about continuous functions, so maybe this just isn't continuous, but what about 1/|x-0.5|. It would rise to infinity at 0.5, then drop back down without the break in the center of the line
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! ❤
r u a math major?
@@Alrightmira I am an associate professor of math education. I wrote this paper on which this is based (to a large extent?) as my master's thesis from UC Berkeley.
@@manyapajama that’s so cool!
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
I've had this experience a lot with math and math CZcams. It's really a consequence of the fact that in school, doing the math is paramount. If you can do the math, it doesn't matter so much whether you have any good intuition for the concepts. Not saying that's a good thing, but it's definitely the principle many schools operate under. (Also, had you not done so much math involving compactness, would this intuitive revelation have really been such a revelation?)
I've seen it in a few classes and had no idea what it really was outside the formal definition
OP Agreed!
@@kruksog I think you're absolutely right that if not for hours spent wrestling the definition into proofs that felt obscene, this video wouldn't have been so powerful. I guess I should really say I'm so lucky to be learning in a time when we have youtubers like morphocular to augment normal education!
Open or close margins?
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 when I first heard the definition, my reaction was "wtf is this thing, and why is this random property interesting?" It wasn't long before it was proven that these sets have magical properties... but I still didn't have much of an intuition for why aside from all the gears magically fitting into place.
I don't think compactness is an abstruse concept at all.
@@pierrecurie man, closed intervals are different from open intervals. That's the motivation. Then you can extract intuition from closed intervals.
@samueldeandrade8535 I kind of understood it from the get go. But that is because before we had topology we had real analysis. And the definition in a metric space is very intuitive(bounded and closed space).
In a general topological space, the definition gets a little dicey for your intuition. But when it clicks you realise how clever the cover-subcover definition is. It's like the epsilon delta click.
@@pierrecuriecompactness is topological finiteness
Back in university, a professor told me that compactness was an analysis/topology-friendly generalisation of finiteness. That has stuck with me since.
There is much truth to this. Basically every theorem where compactness is required is a generalization of a property of finite sets, including the example in the video.
I think this is a very good way to think about compactness. One very useful conceptualization I've come across is to think of finiteness as being a property that is made up of two smaller concepts: discreteness and compactness.
Happy to see Topological concepts being explained so visually and intuitively :)
Ha, that sounds contradictory 😂
MOM NOT RIGHT NOW! I'M BUSY, NEW MORPHOCULAR VIDEO JUST DROPPED
Yeah, yeah, yeah!!! True, true, very true!)
Math God has chosen the moment
content is golden but his voice... damn why is it so irritating🙁
facts fr fr
@@FloppaTheBasedyeah, I feel horrible for saying this, but his voice is the epitome of 🤓
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.
You say "A topological space is compact iff every bounded net has a convergent subnet." This is correct if you drop the word "bounded". In fact, boundedness is not even defined in general topological spaces.
@@twwc960Thanks for the correction, yep I mistyped :)
≠ key exists
@@jeffbrownstain You have an unusual keyboard. I actually prefer the composition of `!=`, `!
@@twwc960Yes! Only in bornological spaces, if I am not mistaken.
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 :>
So glad to hear that! That's what I hoped to do with this video. Best of luck with your studies :)
truly a compact guide
@@morphocular Love your content! But there is something that confuses me. What if the function was f(x)=c or f(x)=sin(4*pi*x/(b-a))? Are we searching for a "global maxima" in the interval [a;b] or just a local maxima.
You have got to have one of the best math channels on CZcams (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.
Wow! Thanks so much for the kind words! It really means a lot to me.
It is the best channel for me.
Babe not now, a new morphocular video just dropped.
After 3 years I have finally understood what "compact" means. Thank you
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".
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.
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
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) 🙂
You can in fact show that continuous functions preserve compactness for any topological spaces, no subsequences needed :D
Indeed the proof is quite short: Suppose f: X \to Y is continuous and surjective (for ease, if it isnt we can replace Y with f(X)) and X is a compact space. Let U_\lambda be an open cover of Y. Consider the collection of the preimage of each of those U_\lambda 's. This is a open cover of X, and must have a finite subcover, say f^{-1}(U_1), ..., f^{-1}(U_n). Then U_1,...U_n is a finite cover of Y.
But why would that be a "second application"? Isnt that a complete replacement of a proof. And a way stronger one too? Or am i misunderstanding.
wish i could send this video to myself 25 years ago! seriously, thank you for the intuitive explanation with excellent visuals
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!
You may already know this, but in fact your statement that (compactness in finite dimensional euclidean space) (closed and bounded) can be further generalised to: (compactness in finite dimensional normed space) (closed and bounded).
@@robbie979Yeah, you're absolutely right. But those are always isomorphic to R^n, if I remember correctly? Don't ask me how to prove that, though :D
But doesn't "closed and bounded" contradict this video at 4:10 and on which discusses open sets that are compact?
@@Graham_Wideman It doesn't talk about open sets that are compact there :) What it does talk about is how you can cover a compact set with a, let's call it patchwork of little open sets. But this patchwork itself isn't compact. It just covers something compact.
@@efi3825 No doubt you are right, in which case the statement at 4:16 "The important sets to pay attention to regarding compactness are the open sets." is highly misleading.
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
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.
The corner note does mention "clopen" sets :3
@@askyleExcellent! I didn't see it since it flashed by quickly
he doesn't say that they are opposites, but he does say that a closed set includes all of its boundary, when it doesn't in the case of clopen sets
@@leave-a-comment-at-the-door a clopen set _does_ include all of its (empty) boundary tho.
@@askyle eeh, maybe I'm misunderstanding the wikipedia article then. it's pretty late now so I'll look at it later
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!
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
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 😅😉
It's one of those things that are super useful but also mysteriously technical and playing a background role, usually nothing flashy enough to get much attention. Same for for much of point set topology.
Great content, as usual. Please keep making such valuable videos, the quality of your work is amazing.
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.
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
This is possibly the best math video on CZcams. Well done!
I love how you start out with an example of how it is useful
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.
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!
this is so well done. i'm not sure if i know any other words that can describe how good this video is
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 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!
Amazing introduction to compactness. I consider myself already quite familiar with the concept, but this made it even more tangible and visual
congrats on nearly 100k, fully deserved
Wow that came full circle, also I got flash-backs to my intro to differential calculus class haha. Great video mate!
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)
i knew nothing about compactness, topology and next to nothing about analysis. Now I understand. Thank you.
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.
Thanks so much! Glad it was helpful!
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
Brilliant explanation of one of Real Analysis' most challenging concepts. Well done!
COMPACTNESS looks like a cool new SNES title the way you formatted the concept.
Very clearly explained as always, Morph!
Thanks for the clear exposition!
Great work on this, thank you.
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.
Really well made video.
Props
It's been many years since I took Analysis but this is the first time I've understood the motivation behind compactness.
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.
Best video I've ever seen in my entire life. I love you
Congratulations on reaching 100K subscribers!
Keep the topology videos coming! They are awesome
Literally studying topology rn. This is perfect timing 🎉🎉
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.
When you use the letter n you still mean reals, right?
Awesome, thank you!
@@Galinaceo0 yes, ill make it more clear
@@user-dk1nr3tv8b If this was the more clear version.... I don't even understand what you're trying to say, let alone how you got to saying it. You had me, tho, all the way up until the 3rd paragraph.
@@kindlin Have you taken a real analysis or topology class? Not that you necessarily would need it but there's a bit of assumed background. If you like I can reply with some "get there as fast as possible" ideas for this specific proof
I never really understood this concept when I took Real Analysis. Thanks for making it clear to me!
Hadn't even heard of compactness before. Thank you for the video.
Amazing video, thanks!
This is a perfect video and it definitely achieves its purpose
Excellent and intuitive explanation :)
Just learned Bolzano Weierstrass in our Real Analysis class, this is really helpful! Thanks!
awesome visualization!!!
I love your videos! Thanks for that!
this is actually making me start to understand some stuff we were just supposed to assume in my intro linear algebra
Thank you so much for making this. Love you😘😘😘😘😘😘
One of the things that drives me crazy about higher math education goes something like this:
I've encountered all of these thought experiments and challenges in classes before but they've never told me why I'm learning them or what use they have to people above me. It's not until I read about something I'm not familiar with or listen to a random CZcams video that I recognize why they were asking me these questions in the first place.
Like, I get that I should be trying to figure them out but I can't go 10 weeks with 3 classes 4 days/wk all exploring results experimentally. Same with novel proofs. I only have so much time.
it is really frustrating in a way that I was able to pass classes like real analysis without ever really grasping the concepts intuitively
Oh my gosh! This is such a good vid! Way better job describing compactness than my advanced calc teacher lol
One of the most beautiful and useful concept in analysis
Great video as always :)
Beautifully explained
So great. Thank you. 🎉
More videos like this, please! To illustrate important math concepts.
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.
That's the essence of maths to me. Intuitive understanding.
Man, really cool explanation
Your way of explaining mathematics is compact , literally
Excellent explanation! 😊🎉
This is a masterpiece! Thank you!
Brilliant video!
EVT proof at the end was great, thanks.
This was straight up better than my Real Analysis and Topology classes
Wonderful video. I loved it.
So much clearer than my analysis teacher!
The transition at 11:21 is just so smooth😊
Extreme Value this channel is
For humanity's sake, you need to make more videos like this one. Great job! I applaud you,
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...
One of the best math channels in the galaxy
An unbounded sequence can definitely have convergent subsequence, and chaotic sequence may neither converge nor diverge, yet not contain any cycle and thus would have no convergent subsequence.
Brings back the memories!
Great video 📷📸 thank you 💯😁
your videos are so so good
you are singlehandedly saving my advanced calc grade
Man this is beautiful. Thanks.
Damn man, that was a brilliant ad
This video explained to me why ford circles are important for N, and I am quite happy about that.
thanks, that really helps with the exam I have in two months
Can't wait to have a deeper grasp on compactness.
Sir your video just earned you a subscriber
This is a great video!