Math Is Fun Forum

ziggs

Member

Registered: 2009-02-22

Posts: 7

Paper on Compact Sets

Hi guys, Was just hoping I could get a little insight from everyone :-)
I'm writing a paper on compact sets and have found a lot of useful theorems and definitions and such,
but need some more interesting or other useful stuff in there.
Was wondering if anyone knew some other theorems or maybe even real life things that rely on the compact sets?
Thanks for any help in advance!

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: Paper on Compact Sets

For some applications in computer sciences you can read some papers like:

http://portal.acm.org/citation.cfm?id=1352928.1353015

http://www.springerlink.com/content/q06j6u2578j78742/

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Paper on Compact Sets

At least in the 2nd paper, they are talking about "compressed" data, an entirely different concept.  I'd have to actually read the first to know about that one, but I have a feeling it is again a different use of the word compact.

ziggs, it would help us if you said what you got so far.  Remember we have no idea what level you're at.  I'm just going to list a couple of things, let me know if you want to know more about them.

For most geometry/topology/analysis applications, compactness lets you go from a local property to a global property.
It can also be seen as a finiteness condition.
While the definition is rather abstract, the Heine-Borel theorem tells you that compactness is very easy in Euclidean space: a set is compact and only if it is closed and bounded.
There are various ways to rephrase compactness in terms of sequences (every set of infinite terms has a limit point, every sequence has a convergent subsequence whose limit is contained in K)
Some related concepts are 2nd countable, paracompactness, locally compact spaces.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

ziggs

Member

Registered: 2009-02-22

Posts: 7

Re: Paper on Compact Sets

First off thanks for responding much appreciated. This is for a real analysis course. I have the Heine-Borel theorem and the theorem's within it. And a few relating to nested sequences and continuity over compact sets. The continuity seems more important. I also see that the Cantor Set is an example. The topic is on Compact Sets, I'm not sure how far i'm allowed to drift off.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Paper on Compact Sets

Ok, that's what I figured.  You probably won't see compact sets really used for a few months (with sequences of functions).  But you can still get an idea about the usage.

Let X be a set, and let P be some property that holds locally.  That is, for every point x there exists a delta such that

Well that's great, but we don't want to have our delta varying as our point x does.  We want to be able to specify one delta for all x (this is like the concept of uniform continuity, if you're familiar).  So we cover our set with these "delta balls".  That is, for each x in X, let delta_x be the corresponding delta from above, and define

So for each point x, we have this delta ball centered at x where the property P holds throughout.  As I said before we're trying to cover X, so the next logical thing to do is to take a union.

Now here is where compactness comes in.  We have an open cover of X, it has to have a finite subcover.  That is,

Where each U_i is really one of the delta balls U_x.  Now for each i, we have the delta_i which is the radius of this ball.  But there are only a finite number of i's.  So we can take a minimum:

Now here is the big punch line.  We went from:

To

Now once we pick a single delta, it works for the entire set X.  This is much better than having to find a different delta for each x in X.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: Paper on Compact Sets

Hi ziggs,

I would like to point out something called compactification
(of non-compact sets). You could google search and look
for the most general topological definition. A subset S of
a set T is compact with respect to a topology on T if every
open cover (collection of open sets whose union contains S)
of S has a finite subcover (finite sub-collection whose union
already contains S).

Now something concrete. We learned at an early age we cannot
divide by 0! That is: 1/0 is NOT defined. Well we could say 1/0
is not a real number as we know. The extended real number
system [-oo, +oo] (only the topological part) is a 2-point
compactification of (-oo,+oo), by adding 2 points, -oo and +oo.
We can define 1/0 to be either -oo or +oo to get a function f
from [-oo,+oo] to itself, with -oo, +oo going to 0.
However f is not continuous.

For continuity of f(x)=1/x, we would like the 1-point compactification,
or further identifying -oo and +oo as a number oo. Define 1/0=oo
and 1/oo=0. Then f is a continuous bijection (homeomorphism) from
(-oo,+oo)U{oo} onto itself. This 1-point compactification of (-oo,+oo)
can be viewed as a circle (but its topology is not the subspace topology of
the real plane). The 1-point compactification of the real plane can be
viewed as a spherical surface (but its topology is not the subspace
topology of the 3-D real space).