Is the Standard Proof of [a,b] in R Being Compact Flawed?

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In summary, the conversation discusses a bone of contention with the standard proof of the closed interval being compact in R with respect to the usual topology. The proof involves dividing the interval in two and claiming that one of the halves is not coverable by a finite subcollection of any open cover. The discussion then addresses the possibility of both halves being uncoverable and the relevance of "picking" in the proof. The speaker suggests that the proof likely involves constructing a nested sequence of uncoverable intervals with decreasing lengths and using an indirect argument. The key factor in each step is the selection of the "uncoverable" half or either one if both halves are uncoverable.
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ice109
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I have a bone to pick with the standard proof of the closed interval in R being compact with respect to the usual topology.

The proof starts out claiming that we can divide the interval in two and it is one of these two halves that is not coverable a finite subcollection of any open cover. Then we proceed to cut that interval in half and make the same claim and so on and so on.

The bone that I pick is what if both halves of the initial interval are uncoverable or both quarters of the initial half and so on.
 
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If both halves are uncoverable at any step, pick either one.
 
  • #3
then why even have any discussion about "picking."?
 
  • #4
I don't know exactly what proof you are using but I'm guessing that you are showing you can construct a nested sequence of uncoverable intervals whose lengths go to zero, probably looking for a contradiction in an indirect argument. So you just have to say how you do each step. You always pick the "uncoverable" half or, if both halves are uncoverable, pick either one because it doesn't matter.
 

1. What does it mean for a set to be compact in the context of real numbers?

In the context of real numbers, a set [a,b] is considered compact if it is both closed and bounded. This means that the set contains its endpoints (a and b) and all values between them. It also means that the set has a finite upper and lower bound.

2. How does compactness relate to continuity in real numbers?

In real numbers, a set [a,b] is compact if and only if every continuous function defined on that set is bounded. This is known as the Heine-Borel theorem. Essentially, compactness guarantees that a function will have both a maximum and minimum value on the set [a,b].

3. Can a set be compact in some contexts but not in others?

Yes, a set can be compact in some contexts but not in others. For example, the interval [0,1] is compact in the context of real numbers, but it is not compact in the context of the set of rational numbers. This is because the set of rational numbers is not complete, meaning it has gaps and therefore cannot contain all of its limit points.

4. How is compactness related to the Bolzano-Weierstrass theorem?

The Bolzano-Weierstrass theorem states that any bounded sequence in real numbers has a convergent subsequence. This theorem is closely related to compactness, as a set [a,b] is compact if and only if every sequence in the set has a convergent subsequence. In other words, compactness guarantees that a set will have enough limit points for any sequence to have a convergent subsequence.

5. Are there any other equivalent definitions of compactness in real numbers?

Yes, there are other equivalent definitions of compactness in real numbers. One is the sequential compactness definition, which states that a set [a,b] is compact if and only if every sequence in the set has a convergent subsequence. Another definition is the open cover definition, which states that a set [a,b] is compact if and only if every open cover of the set has a finite subcover. These definitions are all equivalent and can be used interchangeably to describe compactness in real numbers.

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