Sun 5 Jul 2009

## Compactness

Posted by **David Dudley Field '25** under **Faculty** at 6:14 am

Professor Frank Morgan, in a naked attempt to create controversy and drive traffic, writes:

In my opinion, compactness is the most important concept in mathematics. Here’s an article, recently published in Pro Mathematica, that tracks compactness from the one-dimensional real line in calculus to infinite dimensional spaces of functions and surfaces.

Those are fighting words! Remember when Dan Drezner ’90 used to post pictures of Salma Hayek? Making claims about the most “important concept in mathematics” is the link-generating equivalent among my mathematical friends.

What do you think is the most important concept in mathematics? Being a statistician, I would go with the Central Limit Theorem.

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## 11 Responses to “Compactness”

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Diana says:

David, in a naked attempt to get Diana to comment on EphBlog, posts about mathematical compactness!

One of the more fun parts of my Williams math career was learning four equivalent definitions of compactness — three in real analysis (you only get two if you take applied real analysis!) and the fourth in topology.

(You can think of compactness as meaning, roughly speaking, that you can hold it in your hand — it doesn’t stretch out to infinity in any direction.)

One of my favorite things in mathematics is “one-point compactification,” wherein you take the plane, which is not a compact space, and you

addone point, and suddenly itiscompact! Seemingly, you take something and make itbigger(because you add a point), and then it is somehowsmaller!The central limit theorem is probably a lovely part of statistics, but I didn’t encounter it much in my math career…

July 6th, 2009 at 7:56 pmnuts says:

Compactness for the humanities.

July 6th, 2009 at 9:19 pmlgeorge says:

Brilliant photo link, nuts. And I think I even saw Russia.

July 6th, 2009 at 9:33 pmdm '10 says:

Diana – that intuitive definition of compactness seems more like boundedness to me, which leaves out half of the story. To be compact, a set has to be bounded and

closed: it can’t stretch off to infinity (the set of all real numbers isn’t compact), but it also can’t have any infinitesimal “holes”. Thus the set of real numbers between 0 and 1 but excluding 0.5 isn’t compact, even though it’s clearly bounded, because it omits 0.5 while containing 0.49, 0.499, 0.4999, etc.Also, I don’t think this is quite the type of compactification you were referring to, but that last example does provide a very simple and non-paradoxical situation in which a set can be made compact by adding a single point, since if we add the point at 0.5 then the resulting set [0,1] is clearly compact.

July 6th, 2009 at 9:56 pmJeffZ says:

I think Frank Uible should author a thread to beat up this thread.

July 7th, 2009 at 7:09 amDiana says:

DM ’01: Perhaps I should have mentioned the closed-ness as well, but nothing in the real world is “open” in the mathematical sense of missing just one point. What a strange concept, to be missing one point. It feels like being on the edge of a cliff, like there is an edge there, a strange edge, perhaps quicksand or a vacuum that will suck you in if you get too close.

For the record, we have three Real Analysis equivalent conditions:

– (In R^n): Closed and bounded;

– Every sequence has a convergent subsequence;

– Every open subcover has a finite subcover.

Frank Morgan’s

July 7th, 2009 at 10:28 amReal Analysisis unmatched, in my opinion, at teaching about compactness. Highly recommended. I keep it with me wherever I go.Parent '12 says:

Ah, Diana, I always appreciate your photos. And, I loved your travelogue of finding the intersection of Vermont, Massachusetts, & NY.

But, I’m surprised that I’m enjoyed your addition to the compactness discussion. You presented terms that I actually remember when I decided not to be a math major. And, you got me curious about subcovers!

July 7th, 2009 at 11:55 amDiana says:

Thanks, Parent ’12. I hope to contribute positively to any mathematical discussion on EphBlog. By the way, on the subject of subcovers, open covers in various colors cover the cover of Morgan’s

July 7th, 2009 at 7:45 pmReal Analysis.dm '10 says:

Diana: couldn’t you just as well say that nothing in the real world stretches off to infinity? I think it all depends on what you consider to be part of the “real world”. You probably can’t find infinitesimal holes in real physical objects, but you also probably can’t find any infinitely large physical objects. If you allow certain types of measurements, though, you can find both infinitesimals and infinities. Infinitesimals occur pretty much anywhere where there’s an asymptotic bound that can’t actually be obtained.

For example, the set of “possible speeds of physical objects” is bounded by 0 on the low end and the speed of light on the high end, but it’s not closed (and thus not compact) because it’s impossible to reach the full speed of light. Similarly, if you consider the sets “possible Kelvin temperatures of physical objects” and “possible apparent distances of a physical object from the event horizon of a black hole, from the perspective of an external observer”, neither is closed because both are bounded below by zero, but that zero is impossible to obtain (at least given what very little I know about physics).

July 7th, 2009 at 8:35 pmParent '12 says:

Diana, you remind me that I miss the humor of many who pursue mathematics. I’ll encourage my son to take real analysis.

July 7th, 2009 at 9:47 pmRonit says:

@Parent ‘12: IT’S A TRAP. They reel you in with their humor and infectious enthusiasm for math, and then you end up ruining your GPA. As someone who was once inspired by Diana to entertain thoughts of being a math major, I assure you that I speak from personal experience.

July 7th, 2009 at 10:07 pm