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For each $x \in\mathbb{R}$ let $r_{x} > 0$. Suppose $r_x$ is unspecified and can be different with each $x$. Consider the uncountable family of open intervals:

$U = \{I_{r_x}(x): x \in\mathbb{R}\}$.

The interval $[7, 9]$ is covered by the intervals of $U$. Explain why there exists a finite number of the intervals of U which cover $[7, 9]$. What are those intervals, specifically?

EDIT: My solution: Suppose [7,9] is covered. Since [7,9] is closed and bounded, by the Heine-Borel Theorem, it follows that it is compact. Thus for any open cover, there exists a finite sub cover.

Remy
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But the explicit assumption started with is that $[7,9]$ is covered by the intervals in $U.$ So if all had $x=100$ and $r_x=1$ they would not cover $[7,9],$ against the initial assumption.

coffeemath
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  • Ah, okay. But what if the radius was only .01 each time? How do you know it'll cover with a finite amount of intervals? – Remy Oct 17 '16 at 18:59
  • @JohnH I suggest you look up or google "compact interval" or maybe "compact set". The point is one is assuming the initial collection covers $[7,9]$ so even with small radii and uncountably many centers, the meaning of compact is that there must be a finite subcollection which covers $[7,9].$ This is of course useless unless one already knows that a closed bounded interval in the reals is compact. – coffeemath Oct 17 '16 at 19:02
  • Oh, so the definition of compactness is that for all open covers, there exists a finite sub cover. Since it is assumed that it is compact, there exists a finite sub cover. Is that a sufficient explanation/proof? – Remy Oct 17 '16 at 19:04
  • @JohnH Well depending on what the course is about, one may or may not be allowed to assume closed bounded intervals are compact. If not, one would have to essentially prove that in answering. – coffeemath Oct 17 '16 at 19:06
  • I have learned about the Heine-Borel theorem, so I think I can. Made a change to my solution above. – Remy Oct 17 '16 at 19:15
  • @JohnH OK, if you are allowed to use Heine-Borel, then all you need to do is show is that $U$ covers $[7,9]$. – arkeet Oct 17 '16 at 19:22
  • Would that be difficult to do? I feel it may be assumed in the question but not sure. – Remy Oct 17 '16 at 19:24
  • @JohnH If one can use that the reals satisfy the least upper bound property, then a proof of compactness of a bounded closed interval is not very difficult. Note that there has to be some assumption about the reals to prove the statement in your question. – coffeemath Oct 18 '16 at 04:06
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Compactness guarantees you only need a finite number out of the collection to cover the closed interval. You know the interval is covered because for each $x$, $x\in I_{r_x}$.

Unless you're given some of the values of $r_x$ for a few given $x$, you can't really say which ones cover the interval specifically

snulty
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  • But how do you prove the interval is compact? This seems circular. – arkeet Oct 17 '16 at 19:08
  • @arkeet To prove a closed interval is compact you need to do some analysis. This involves using the completeness of the reals. I can probably find and link you a proof if you like. – snulty Oct 17 '16 at 19:10
  • @snulty Heine-Borel? Is that the theorem this refers to? – Remy Oct 17 '16 at 19:12
  • @arkeet Here's one on mse. – snulty Oct 17 '16 at 19:15
  • @snulty Right. So essentially the question asks to prove Heine-Borel for $\mathbb{R}$. – arkeet Oct 17 '16 at 19:16
  • @snulty Could you take a look at my edited proof above please? Does that suffice? – Remy Oct 17 '16 at 19:16
  • Also, I'm not sure how to specifically state the intervals, because like you said, that can't really be done. But maybe there is a general way or something? – Remy Oct 17 '16 at 19:18
  • @arkeet I'll be honest, I'm not sure what's explicitly asked. If you think that showing the function $x^2-2$ has a zero involves showing that the function is continuous and satisfies the intermediate value theorem then I agree with you. Otherwise I think it should be stated what can be assumed true etc. – snulty Oct 17 '16 at 19:20
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    @JohnH if it's for a course, maybe they've said what you can assume, for example the heine borel. If it's from a book, then whatever precedes the question I'd say is fair game (although I can think of book(s) which is an exception). If the $r_x$'s are tiny there could be hundreds of intervals, I'm not sure how to estimate a bound on the possible number of intervals needed. I'll think about it now though. – snulty Oct 17 '16 at 19:24
  • I think rather than giving specific intervals, I should put no, you can't give specific intervals – Remy Oct 17 '16 at 20:05
  • @JohnH Yeah at the moment I'm still not sure what you can say specifically you can say about the intervals. – snulty Oct 17 '16 at 21:49