Why is compactness important in topology?

[Tac-Tics]

Hi all.

I recently picked one of my favorite math books out of my closet and started flipping through it again (Gamelin & Greene's Intro to Topology).

I'm continuing through the second half now, which is on algebraic topology. But there was one thing that I never quite "got" in point-set topology, and that's the intuition behind compactness.

I understand the definition. A compact set is a set where every cover has a finite subcover. I understand it's equivalent to being closed and bounded in R^n. But what I don't quite get it why its important.

I've heard the saying "compactness is the next best thing to finiteness". It sounds like this hints at an underlying motivation for the property. Maybe it might help to elaborate on this quote.

Sorry for the unruly open-ended question =-)

 

[VKint]

I don't know a lot of topology, but the most important reason for considering compact spaces that I know of is that, in a compact space, all sequences of points have convergent subsequences. (Certainly this is the reason that compactness comes up so often in analysis.) In this context, the statement "compactness is the next best thing to finiteness" could be alternatively rendered as "a compact space is 'sufficiently finite' that any infinite sequence of points within it must 'accumulate' at at least one point in the space." Eventually, that is to say, if you visit infinitely many points in the space, you run out of room and are forced to return arbitrarily close to somewhere you've already been.

 

[rasmhop]

I must admit I don't think I have a complete conceptual grasp of what it means for a space to be compact, but I will try to share how I intuitively feel about compact spaces.

For closed sets I think of it as a kind of abstract analogue to boundedness. Sets that are not closed cannot be compact if they are Hausdorff, so this covers all cases for Hausdorff spaces. For non-Hausdorff spaces I don't really have a good way of thinking about them except as in terms of open covers or convergent subnets (a space is compact if and only if every net in it has a convergent subnet). So I see it as a way of stating both that a net can't keep getting further away from the previous points, at some point it needs to return to the same general area, but as nets can also be a bit hard to visualize I'm not sure how much this helps.

I also like the next-best thing to finiteness idea. In topology we can both deal with points and open sets, and while we may sometimes prefer to deal with points and consider open sets as collections of sets in many cases it's much more natural to consider open sets as the most fundamental concept. Compactness is a kind of finiteness, just for collections of open sets instead of collections points. Also finiteness of open sets often turns out to be exactly what we need as we can then take both intersections and unions and preserve openness.

I guess the main reason that we use compactness is not that it is such a terrible natural concept, but rather that we have found it to be a concept that is close to the minimal thing we need to assume to show a number of interesting theorems and it also turns out to be well-behaved in many respects (closed subsets preserve the property, products preserve the property, continuous maps preserve the property). Concepts like sequential compactness are much easier to think about, but they turn out to be less useful than compactness.

 

[LeonhardEuler]

It's been a while since I've studied topology, but I mainly remember thinking of compact sets as being a way of generalizing and abstracting the properties of closed and bounded sets of Euclidean spaces to more general contexts, where there may not even be a metric. I remember my professor saying that it was not clear historically how this should be done, but the idea of "compactness" caught on just because it is what turned out that important theorems could be proven with that condition, and it was still fairly general.

 

[VeeEight]

Perhaps reviewing theorems that involve compact sets like Ascoli Arzela, Stone Weierstrass, etc will also help you conquer compactness.

 

[Hurkyl]

One commonly useful fact about compactness is the image (under a continuous function) of a compact set is compact.



I think rasmhop's "next-best thing to finiteness" paragraph describes well the overarching Big Idea^TM behind compactness. All over mathematics, one often tends to study a complicated object by trying to describe it in terms of simpler objects.


Suppose, for example, you want to prove that every continuous function on the sphere is bounded.

Well, the sphere can sometimes be hard to visualize, and continuous functions can very often be very strange and hard to visualize. So, I need some way to tame the problem so I can make sense of it.

Well, I'm interested in boundedness, so I'll try to do something in terms of that! I choose some point which I will call P0, and I define an open set U0 to consist of all points on the sphere such that $\mid f(P_0) - f(x)\mid < 1$.

Okay, that's good -- I know f is bounded on U0. Let's try again -- let's pick some point P1 in U0, and define U1 to be the set of points so that $\mid f(P_1) - f(x) \mid < 1$. Now, I know f is bounded on the union $U_0 \cup U_1$, because everything can get back to P0 in 2 "steps", on each step, f varies by at most 1.

So I might think we can keep going -- but now I'm worried: what if this process never ends? What if there are points on the sphere that aren't in any of my U's?

Well, I realize the sequence doesn't matter -- just that I can get from any point back to P0 in finitely many steps. So I will take an open set Ua for every point a, defined as the set of points such that $|f(a) - f(x)| < 1$.

I know the sphere is connected, and that tells me^* that any point can get back to P0 in finitely many steps... but I have no guarantee that the number of steps is bounded!
*: edit: Now that I think about it, while this statement seems obvious, I suspect the proof is not trivial -- I'm worried there may even be a counterexample! (with some space other than the sphere, of course)

Aha! Now I invoke compactness -- I can extract from my infinite collection a finite subset that still covers the entire sphere. Now, since I have finitely many open sets, I know each point can be connected back to P0 in finitely many steps. Success!


(How did I know the sphere was compact? Was it a hard and difficult theorem? Nope -- I just realized the sphere was the union of two compact sets: specifically two closed disks)

 

[blerg]

The biggest thing about compactness that I have come across is that for a compact set, results that hold locally also hold globally.

For example, local boundedness on a compact set implies global boundedness for the whole set.
Other properties follow the same idea.

 

[Tac-Tics]

Reading through the book and some online lecture notes, I see now that the key component of compactness, as many here have described it, is an abstract "boundedness". That there are no parts which run off towards "infinities" in any sense, but in a metric-free way. That the intuition is centralized on the set being bounded, not closed.

Thank you for everyone who chimed in. And thanks Hurkyl for your detailed example -- I will reflect on that one.

 

[George Jones]

The dozen pages in chapters 29, Nets, and 30, Compactness, of the book Mathematical Physics by Robert geroch are very much worth reading.

 

[haushofer]

I must admit I don't think I have a complete conceptual grasp of what it means for a space to be compact, but I will try to share how I intuitively feel about compact spaces.

For closed sets I think of it as a kind of abstract analogue to boundedness. Sets that are not closed cannot be compact if they are Hausdorff, so this covers all cases for Hausdorff spaces. For non-Hausdorff spaces I don't really have a good way of thinking about them except as in terms of open covers or convergent subnets (a space is compact if and only if every net in it has a convergent subnet). So I see it as a way of stating both that a net can't keep getting further away from the previous points, at some point it needs to return to the same general area, but as nets can also be a bit hard to visualize I'm not sure how much this helps.

I also like the next-best thing to finiteness idea. In topology we can both deal with points and open sets, and while we may sometimes prefer to deal with points and consider open sets as collections of sets in many cases it's much more natural to consider open sets as the most fundamental concept. Compactness is a kind of finiteness, just for collections of open sets instead of collections points. Also finiteness of open sets often turns out to be exactly what we need as we can then take both intersections and unions and preserve openness.

I guess the main reason that we use compactness is not that it is such a terrible natural concept, but rather that we have found it to be a concept that is close to the minimal thing we need to assume to show a number of interesting theorems and it also turns out to be well-behaved in many respects (closed subsets preserve the property, products preserve the property, continuous maps preserve the property). Concepts like sequential compactness are much easier to think about, but they turn out to be less useful than compactness.

I really like this kind of conceptual clearification of rather abstract mathematical notions. I also have pondered about this. Your post demystefies a lot. Thanks!

 

[HallsofIvy]

Compactness is "the next best thing to finite". That is, many of the properties of finite sets (closed, has upper and lower bounds for ordered spaces) are also true of compacts sets precisely because you can always replace an open cover by a finite sub-cover.

 

[morphism]

I like to think of compactness as the topological analogue of finiteness. It's not too hard to justify why: Usually when we attempt to make the passage from finite to infinite in some mathematical theory, we try to use topology to help us tame the infinite. And it's often the case that if the topology is compact, the infinite theory behaves very much like the finite one.

A very nice example of this can be found in representation theory. One of the pillars of the representation theory of finite groups is the happy fact that (complex, finite-dimensional) representations split into direct sums of irreducible subrepresentations. This is an important result because it transforms the problem of studying the (complex, finite-dimensional) representations of a finite group into the apparently more manageable one of studying only the irreducible representations. The infinite-dimensional analogue of this would be the statement that every continuous representation of a topological group (finite or infinite) splits into a direct sum of irreducible subrepresentations. Alas this is not true for general groups, but it is true for compact groups.

 

[g_edgar]

I've heard the saying "compactness is the next best thing to finiteness". It sounds like this hints at an underlying motivation for the property. Maybe it might help to elaborate on this quote.

Very beautifully explained, with interesting examples, in a paper by E. Hewitt in 1960 published in the American Mathematical Monthly

 

[wofsy]

Hi all.

I recently picked one of my favorite math books out of my closet and started flipping through it again (Gamelin & Greene's Intro to Topology).

I'm continuing through the second half now, which is on algebraic topology. But there was one thing that I never quite "got" in point-set topology, and that's the intuition behind compactness.

I understand the definition. A compact set is a set where every cover has a finite subcover. I understand it's equivalent to being closed and bounded in R^n. But what I don't quite get it why its important.

I've heard the saying "compactness is the next best thing to finiteness". It sounds like this hints at an underlying motivation for the property. Maybe it might help to elaborate on this quote.

Sorry for the unruly open-ended question =-)

This does not really answer your question but gives me a lot of intuition about compactness. That is that every discrete subset is finite - the next best thing to finiteness.

Do you think that this condition is equivalent to compactness?

 

[Sina]

Suppose a set A has inifinite elements. You would wish that it would be finite. Compactness brings about this: take very small neighbourhoods U_i around each of your points. The collections of these nbds are infinitely numbered but (if A is compact) among these you can find a finite number of U_j, whose union still covers or "represents" your set. Even if you have not broken your set into finite number of elements, you have broken it into finite number of very small sets. Don't let this example mislead you though as you can do this selection thing for any such open covering, not only the one I have described here.