Heine Borel Theorem, why I need it?

[zli034]

Hello all:

Closed interval subcover is finite. How do I use it? Why should anyone on Earth proved things like this? Please give me the significance of this technological development.

Thanks,

zli034

 

[micromass]

Hi zli034!

I guess you mean to say: every open cover has a finite subcover...

Anyway, the development of compactness has been a long one, and that makes it pretty hard right now to see the significance of some things.

The story begins with analysts like Weierstrass, Cauchy, Bolzano,... They wanted to formalize things like continuity, limits, etc. And some of the most important theorems had to do with closed intervals:

The extreme value theorem: Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function, then f attains a minimum and a maximum.
​

Other nice things we can do on closed intervals is for example defining integrals. Of course, we aren't happy with this, we want to make this more general. Because, what if our domain is 2-dimensional or 3-dimensional? What should be the generalization of closed intervals? Well, it appears that closed+bounded is the right analogon. We have:

The extreme value theorem: Let $X\subseteq \mathbb{R}^n$ be closed and bounded and let $f:X\rightarrow \mathbb{R}$ be a continuous function, then f attains a minimum and a maximum.
​

Everything looks cool now, but things needed to be much more general. Developments in physics and mathematics (for example, differential equations), require us to look at more general things than $\mathbb{r}^n$. These general things are metric and topological spaces. Of course, the terminology "X is closed and bounded" makes sense in metric spaces, but the extreme value theorem is false under that assumptions!

So mathematicians set out to find analogons to "closed and bounded" in metric and topological spaces. And after quite a long search, they found the definition with subcovers. With that definition, we indeed have

The extreme value theorem: Let $X$ be a compact metric space and let $f:X\rightarrow \mathbb{R}$ be a continuous function, then f attains a minimum and a maximum.
​

So our definition of compactness indeed satisfies things like the extreme value theorem, and allow us to define integrals. Other applications are the theorem of Ascoli-Arzela where our new definition of compactness is essential!

Finally, it is of course the question if our notion of compactness agrees with "closed and bounded" for subsets of $\mathbb{R}^n$. This is exactly the theorem of Heine-Borel.

Note that the original definition of compactness was much more complicated. And the definition of open covers is actually a relative simple one...

 

[zli034]

They should call Heine-Borel a lemma, not theorem. Theorem can be used to derive quantities. Lemma is for proving theorems.

 

[HallsofIvy]

I am going to assume that English is not your native language. "Theorem" and "lemma", in English, do NOT mean what you seem to think.

 

[mathwonk]

every continuous function is locally bounded. i.e. each point has a neighborhood on which the function is bounded. Hence if a finite number of such neighborhoods cover a whole interval, then the function is also bounded on the whole interval. see how useful that is?

Heine Borel compactness is a generalization of finiteness, a useful property.

 

[Fredrik]

They should call Heine-Borel a lemma, not theorem. Theorem can be used to derive quantities. Lemma is for proving theorems.

If what you mean by "to derive quantities" is "to calculate stuff in the real world", that view is a bit naive. Mathematics is so much more than just "how to calculate".

I would say that the words "proposition", "theorem", "corollary" and "lemma" all mean the same thing, but are used in slightly different contexts.

Theorem: The word you choose if you don't have a reason not to.
Proposition: Some authors use this for theorems that are easier to prove or of lesser significance than the ones they reserve the word "theorem" for.
Corollary: A theorem that's easy to prove, if you use another theorem that you just proved.
Lemma: A theorem that's not very interesting on its own, but is useful to prove before an interesting theorem, so that the proof of the interesting theorem will be shorter and easier to follow.

 

[Skrew]

If you prove general theorems for functions with a compact domain, and you know that a closed and bounded subset of R^n is compact, you know all your theorems which have been proven apply to functions in R^n over a closed and bounded subset, etc.