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0:00
let's start things off with a puzzle say I give you a bounded segment of the x-axis maybe the open interval from 0 to 1. so all real numbers strictly between 0 and 1 but not including 0 and 1 themselves can you draw the graph of a continuous function that has no finite maximum on that interval that is can you draw a single continuous curve moving from left to right that has no highest y coordinate many of the regular kinds of Curves you could draw wouldn't be able to do this
0:34
but if you've seen enough graphs in your life you might have an idea try the function 1 over X its graph forms a vertical asymptote at x equals zero meaning the height of the graph grows without bound as X approaches zero though the graph never actually reaches x equals zero thus it has no finite maximum alright but what if I require the graph to actually touch both x equals 0 and x equal 1. that is the graph must actually contain a point with x coordinate 0 and
1:06
another point with x coordinate one in addition to all the values in between so no funny business with asymptotes or holes but it still needs to be a single continuous curve can you draw a curve that has no highest point now
1:22
not so easy is it it's tempting to try and fit an asymptote somewhere but anywhere you put one will result in a discontinuity since the function now has to be defined at the actual asymptote Point itself and we're not counting Infinity as a number here so it seems like we're forced to actually have a legitimate highest and lowest point on the graph we can't avoid it but is it actually impossible though how do you know if you've never seen an asymptote before you might not have thought of that
1:51
solution in the previous case so how do you know there's not some similarly clever trick that will allow the function to avoid having a Max or a Min in this case well it's questions like these that the concept of compactness helps solve it's 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
2:26
compactness is a property of shapes or to be more precise of sets of points in some kind of space this time for a change how about we just take a look at the formal definition right off the bat a pretty standard definition of compactness would go something like this a subset K of a topological or metric space is called compact if every open cover of K has a finite subcover all right now let's go and uh unpack this definition first of all it mentions a topological or metric space what is that
3:03
I went over this briefly in my video about connectedness but just to recap remember in topology all shapes curves Etc are thought of as a collection or set of points and any set of points s divides the space it lives in into three other sets the interior of s consisting of all points that can be enclosed in a ball that's also contained in s the exterior of s consisting of all points that can be enclosed in balls that are completely disjoint from s and finally the boundary of s points
3:37
where any ball containing them no matter how small are only ever partially inside or outside s the critical thing to note is that a set may or may not include its boundary or can even include just part of it a set that includes all of its boundary is called closed a set that includes none of its boundary is called open and a set that includes only part of its boundary isn't special enough to get a name so getting back to our definition without going into too much detail a
4:07
topological or metric space is just a space where Concepts like open closed interior exterior boundary and so on makes sense the important sets to pay attention to regarding compactness are the open sets those are sets that only consist of interior points which is important since it means they always have to be a little bit fat since any points inside them have to be able to be enclosed in balls that are also inside the set this brings us to the next part of the
4:38
definition open covers this one is pretty straightforward an open cover for a set is a collection of open sets that collectively cover the target set there could be finitely many of them or infinitely many and can individually have any size a sub cover is just a sub collection of these open sets that still manages to cover the target set note we also include the original cover itself as one of its subcovers so putting all this together a compact set is a special set where for any
5:12
possibly infinite collection of open sets you come up with to cover the set you can always select a finite sub-collection of them that will still cover that set note that this isn't the same as saying the set can be covered by finitely many open sets that's actually true of all sets just pick a big enough open ball to cover it or if it's an unbounded set just use the entire space itself which counts as an open set no this is saying something more subtle it's saying that if you give me a
5:43
potentially crazy complicated collection of open sets that covers the compact set I can always cover the set using only finitely many of the open sets you gave me or to put it another way any infinite collection of open sets that together cover a compact set always over covers it you can always get away with a finite sub-collection and that's it that's all a compact set is not too crazy a concept once you've digested it but even so it's not very clear yet what kind of sets would fit
6:16
this description in fact right out the gate it sounds like they'd have to be pretty strange and complicated right and furthermore why would sets would this property be that interesting or important anyway so let's see if we can get a better intuitive sense of what these sets are and what they can do
6:37
the simplest kind of compact set is just a finite set a collection of finitely many points because if I cover a finite collection of points with open set blobs I can just go Point by point and select one open set from the cover that contains it and in the end I'll have selected at most one set per point though in practice maybe fewer since some of the open sets I selected might cover multiple points this actually gets pretty close to the heart of what compactness is meant to do
7:07
compact sets sort of simulate finite sets for topological purposes in the sense that certain tricks or techniques that work on finite sets also work on compact sets but more on that later Beyond finite sets the first examples of compact sets you're likely to be introduced to are closed and bounded sets of the real number line so the closed interval 0 to 1 we saw at the beginning of the video is an example of a compact set the fact that closed and bounded sets of
7:38
real numbers are compact is known as the hiney borrel theorem and is a foundational result in analysis I won't go through the proof of it here though since it takes a little finesse and I also don't think it sheds that much light on the broader question of how we should intuitively interpret what compactness means for that I think it helps to turn to an alternative description of compactness called sequential compactness to understand it think about a sequence of points
8:06
depending on the sequence it may or may not converge toward a limiting point if it doesn't converge it may fail by diverging off to infinity or by wandering aimlessly around without approaching anything but in the second case you can sometimes extract a convergent sequence out of it by selectively ignoring certain points in the sequence here's an example consider the sequence which just alternates between 0 and 1. this sequence is Divergent since it never makes up its mind so to speak
8:37
about whether it's approaching zero or one but if we selectively ignore every other term we get a sequence that just looks like an endless string of zeros or ones which of course converges to zero or one that's an extreme example but more generally if you have a sequence that diverges but nevertheless forms clusters in certain spots then by carefully throwing out points not in the Clusters you can sometimes extract a convergent sequence out of it what you might call a convergent subsequence
9:12
now bringing it back to compactness it turns out a set is compact if and only if every sequence of points that stays within that set as a subsequence that converges to a point also in that set that is a compact set is a set which forces any sequence of points within it to form clusters that converge toward a point also within the compact set this is called sequential compactness and it's equivalent to the standard notion of compactness except in certain more unusual topological spaces which we
9:45
won't worry about here from this New Perspective we can perhaps get a sense for how to intuitively think about compact sets by studying the different ways a set can fail to be sequentially compact or in other words the different ways to prevent a subsequence from converging the simplest way is just to have the original sequence diverge off to Infinity then any subsequence will also diverge to Infinity meaning any set containing such a sequence won't be compact
10:15
so the first requirement of a compact set is that it should be bounded it can't be infinitely big or elongated the second way to stop a sequence from having a convergent subsequence is to have it approach a whole or a missing point in the set which could either be a gap in the interior of the set or a missing boundary then every subsequence will also approach that hole and will fail to converge so the second requirement for a set to be compact is that it should not have
10:46
any such missing points it should be complete with the two requirements of boundedness and completeness it might seem like we've covered every failure case but there is actually a third very subtle way a sequence can avoid having a convergent subsequence but to see it we'll need an infinite dimensional space that may sound intimidating but there are actually some relatively simple examples think about it this way any point in a two-dimensional space can be described with two numbers X and Y
11:20
likewise a point in a three-dimensional space can be described with three numbers x y and z so a point in an infinite dimensional space can be described with an infinite list of numbers A1 A2 A3 and so on to keep things a little simpler for this example I'll restrict the space to only include lists that eventually end in an endless string of zeros with this in place consider this sequence of points which are actually lists each list contains only a single one
11:52
surrounded by nothing but zeros where the nth list in the sequence has a one in the nth place this sequence of lists is bounded all of them are a distance of one away from the zero list but it doesn't converge this sequence of lists doesn't approach any list you might think it converges to the zero list since any particular slot in the lists eventually becomes a zero but from a strictly topological point of view it doesn't approach the zero list since any list in our sequence stays a distance of
12:26
one away from the zero list and naturally any subsequence will suffer the same fate in effect we have a sequence of points that endlessly explores all the infinitely many coordinate axes of the infinite dimensional space while always staying a fixed distance away from the origin so it avoids approaching anything by kind of traveling through infinitely many dimensions sounds like bad science fiction but it's true mathematics the requirement we'd need to impose on a
12:58
set to prevent this is called total boundedness and it basically says that your set can be covered by a finite number of balls of Any Given fixed size in our familiar finite dimensional euclidean space this is the same as just ordinary boundedness since any bounded blob can always be covered by finitely many balls of any fixed size no matter how small it may take a lot of them if they're really small but it's always a finite number not so for our infinite dimensional set
13:29
though since if I surround each point in our sequence with a ball of radius 1 4 for example all the balls will be disjoint from each other since every list in the sequence is a distance of more than one half away from every other list so the set this sequence of lists lives in would not be totally bounded with total boundedness in place we finally actually got enough to force every sequence in a set to have a convergent subsequence that is to be compact because there's a theorem that
14:00
says a set is compact if and only if it is complete and totally bounded so this is how I like to think about compactness intuitively a set is compact if it's small enough and solid enough that there's no way for a sequence inside it to escape to infinite distance Escape into a hole or Edge or Escape through infinite dimensions in ordinary finite dimensional euclidean space this corresponds precisely to the sets that are closed and bounded but as we saw that's not always enough for more
14:33
exotic spaces so that's how I view compact sets intuitively but even so why are such sets so important to analysis and topology what do they let us do let's look into that now [Music] one of the main reasons compact sets are so nice is they often provide a way to take properties that hold locally on the set and extend them to apply globally on the entire set that may sound a little vague so let's do an example remember the puzzle at the beginning of the video where I challenged you to find
15:10
a continuous function on the closed interval 0 to 1 that didn't have a maximum well it turns out it is indeed impossible and it comes down to the fact that the closed interval 0 to 1 is compact every continuous function defined on a compact set attains both a maximum and a minimum this fact is known as the extreme value theorem and to show how compactness plays a key role we'll prove a slightly weaker version of it that any continuous function defined on a compact set is
15:41
bounded meaning the function's outputs have both a finite upper bound and lower bound but we won't require the function to actually reach those bounds just so we have something concrete to look at let's take that compact set to be the closed interval 0 to 1. to begin consider an arbitrary real number X in the interval 0 to 1. since our function f is continuous it Loosely means small changes in X result in relatively small changes in f of x in fact if I restrict the variability of
16:13
x to a small enough interval call it a Delta interval I can force the function's outputs f of x to stay within certain bounds like I don't know a distance of 0.1 from where it started so in essence I've defined a certain open interval around X where the image of that interval after applying F to it results in a bounded interval in the output space in other words f is bounded locally around X can you see where this is going we can construct one of these little Delta intervals around every single
16:48
point x in the domain creating potentially an uncountably infinite collection of Delta intervals that together cover the domain 0 to 1. but since the domain 0 to 1 is compact it means we've overcovered it and so we can reduce down to a finite sub-collection of these Delta intervals that still cover all of 0 to 1. but by construction the function f is bounded on each of these remaining Delta intervals and since there are only finitely many such Delta intervals there
17:21
must be a finite overall upper Bound for f just take the highest upper Bound for f among all the remaining Delta intervals and likewise we can find a finite overall lower bound thus f is bound to Tire interval 0 to 1. we've taken local boundedness and turned it into global boundedness but notice how this proof falls apart if the domain isn't compact like on the open interval 0 to 1. if we try the same trick here for the function f of x equals 1 over X we'd find that Delta
17:57
intervals for the X values get thinner and thinner the closer we get to x equals zero in order to keep the same local bounds on y since the function is so steep there it doesn't take much variability of x to get y to vary beyond our maximum chosen tolerance and here there'd be no way to reduce to a finite sub-collection of these Delta intervals since we truly need an infinite number of them in order to cover the interval all the way down to x equals zero since they'd be getting
18:28
thinner the closer we get and if there are infinitely many Delta intervals we can't guarantee that there is a single Delta interval that yields the highest possible upper Bound for the function f and for the function 1 over X specifically there definitely isn't one that's an overview of what compactness is and why it's so important and useful indeed you'll encounter the concept over and over again in higher math where it tends to be an important ingredient in
18:57
enabling many powerful constructs such as function spaces where functions can be treated like infinite dimensional vectors and studied as such According to some being able to understand compactness might even make the difference in being able to participate effectively in higher math much of modern math is built on compactness and if that's true I hope this video has made it a little more accessible but speaking of making Math More accessible one of the best resources
19:26
I've encountered for providing a thorough and enjoyable way to learn math is this video sponsor brilliant.org one of the main applications of compactness is in the rigorous study of probability and if you need a way to learn or brush up on your probability fundamentals at your own pace brilliant offers several courses on It featuring their signature interactive lessons like their intro to probability course which I like a lot as they have you build and run simulations right in the
19:52
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20:20
trial by heading over to brilliant.org more vocular and the first 200 of you to sign up will get 20 off in annual premium subscription so thanks to brilliant for sponsoring this video and I'll see you in the next one
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