Understanding Compact Spaces: A Beginner's Guide

[lavster]

Hi,

can someone explain to me what is meant by a compact space? I don't understand the definitions on the web... my knowledge of alegbra is neglible...

thanks

 

[klackity]

Well, compactness is a topological notion, not an algebraic one.

Here's the formal definition:

A topological space is compact if every open cover has a finite subcover.​

In order to understand this definition, you need to know the definitions of "open", "cover", and "finite subcover". I'll assume you know what open means, but correct me if I'm mistaken.

A cover of a space is a collection of sets whose union contains the whole space.​

A finite subcover of a cover is a finite subcollection of the original collection which still covers the whole space​

The intuitive idea behind compactness is that things don't really have "anywhere to go" in a compact space.

 

[HallsofIvy]

For example the set {1, 1/2, 1/3, 1/4, ...}= {1/n}[/tex] is NOT compact because there exist an &quot;open cover&quot; that has no &quot;finite subcover&quot;. For each of the numbers in the set, let the open interval from half way between that number and the previous number to half way between that number and next number (for the first number, 1, which has no &quot;previous&quot; number, take a number larger than 1 instead). Every number in the set is in such an interval- and in only 1. That is, 1 is in the interval from 2 to 3/4, 1/2 is in the interval from 3/4 to (1/2+ 1/3)/2= 5/12, 1/3 is in the interval from 5/12 to (1/3+ 1/4)/2= 7/24, etc. Since every number in the set is in such an open interval this is and &quot;open cover&quot;. Since every number is in only one of them, we can not drop any such interval and so can not have a &quot;finite subcover&quot;.<br /> <br /> On the other hand, if we add the number &quot;0&quot; to that set, we get a compact set. That is because the numbers 1, 1/2, 1/3, ..., 1/n... converge to 0. Given <b>any</b> collection of open sets that includes all of those numbers, one of them must include 0. But then it must include some small interval of 0 (a property of open sets) and there exist some N such that is n&gt; N, 1/n is in that interval. That leaves only a finite number of terms (n\le N) so we can choose one set containing each and have a &quot;finite subcover&quot;. This is an example of the &quot;Heine-Borel theorem&quot; that compact sets in the real numbers are compact if and only if the set is both closed and bounded.<br /> <br /> Another way of looking at it is that compact sets are &quot;the next best thing&quot; to finite sets. Compact sets have many of the properties of finite sets- they are closed and bounded, ordered compact sets always have a largest and smallest member, etc.

 

[micromass]

Hi lavster!

I thought it could be useful if I gave some motivation about the notion of "compactness". In analysis, there are a few important theorems, one of them is the extreme value theorem.
As you probably know, this states that

Any continuous function f:[a,b]\rightarrow \mathbb{R} attains a minimum and a maximum.

This theorem is extremely useful as it can be used to show Rolle's theorem, Taylor's theorem and several others. So mathematicians began to think whether they could probably change the domain [a,b] to a more general one (why? because the theorem could have useful generalization to multivariate analysis!). And this was indeed possible! We can generalize the theorem as follows:

Let X\subseteq \mathbb{R}^n be closed and bounded. Then any continuous function f:X\rightarrow \mathbb{R} attains a minimum and a maximum.

Still, mathematicians were not happy. Indeed, several developments in mathematics and physics lead them to consider metric spaces and topological spaces. These are concepts in which notions such as continuousness and convergence still make sense.

The question is whether we could generalize the extreme value theorem to the setting of metric spaces and topological spaces. This is indeed possible and it leads us to the notion of compactness. In metric spaces, a space is compact if every sequence has a convergent subsequence, or equivalently, if every open cover has a finite subcover. This is the correct generalization, since we have

Let X be compact and let f:X\rightarrow \mathbb{R} be a continuous function, then f attains a minimum and a maximum.

Furthermore, this notion of compactness specializes to closed and boundedness for subsets of \mathbb{R}^n (this is the Heine-Borel theorem). Thus this definition is really what we're looking for!

For topological spaces, the situation becomes slightly more complicated as the statements "every sequence has a convergent subsequence" and "every cover has a finite subcover" are no longer equivalent. It was discovered, however, that the latter was the proper generalization. The sequence-statement was less important, but still got the name sequential compactness.

Also note that the history of compactness was a very long one. Right now, we can give relatively easy definitions of compactness, but the original definitions were quite complicated!

 

[chiro]

Hi lavster!

I thought it could be useful if I gave some motivation about the notion of "compactness". In analysis, there are a few important theorems, one of them is the extreme value theorem.
As you probably know, this states that



This theorem is extremely useful as it can be used to show Rolle's theorem, Taylor's theorem and several others. So mathematicians began to think whether they could probably change the domain [a,b] to a more general one (why? because the theorem could have useful generalization to multivariate analysis!). And this was indeed possible! We can generalize the theorem as follows:



Still, mathematicians were not happy. Indeed, several developments in mathematics and physics lead them to consider metric spaces and topological spaces. These are concepts in which notions such as continuousness and convergence still make sense.

The question is whether we could generalize the extreme value theorem to the setting of metric spaces and topological spaces. This is indeed possible and it leads us to the notion of compactness. In metric spaces, a space is compact if every sequence has a convergent subsequence, or equivalently, if every open cover has a finite subcover. This is the correct generalization, since we have



Furthermore, this notion of compactness specializes to closed and boundedness for subsets of \mathbb{R}^n (this is the Heine-Borel theorem). Thus this definition is really what we're looking for!

For topological spaces, the situation becomes slightly more complicated as the statements "every sequence has a convergent subsequence" and "every cover has a finite subcover" are no longer equivalent. It was discovered, however, that the latter was the proper generalization. The sequence-statement was less important, but still got the name sequential compactness.

Also note that the history of compactness was a very long one. Right now, we can give relatively easy definitions of compactness, but the original definitions were quite complicated!

Thank you for that micromass, it was very informative.