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Chapter 4: Problem 3

Show directly that the interval \([0, \infty)\) does not have the Heine-Borel property.

Short Answer

Expert verified
The interval \([0, \infty)\) is not bounded, so it lacks the Heine-Borel property.

Step by step solution

01

Understanding Heine-Borel Property

The Heine-Borel property states that every open cover of a set has a finite subcover if and only if the set is closed and bounded. Here, the question is asking us to show that the interval \([0, \infty)\) does not satisfy this property.
02

Define an Open Cover of the Interval

Consider the collection of open intervals \( \{ (0, n) \,|\, n \in \mathbb{N} \} \). Each interval \((0, n)\) covers the interval \([0, \infty)\), as for any point \(x\) in \([0, \infty)\), there is a natural number \(n\) such that \(x < n\).
03

Check for a Finite Subcover

To have a finite subcover, there must exist finitely many intervals from our collection that together still cover \([0, \infty)\). However, any finite subcollection \(\{ (0, n_1), (0, n_2), \ldots, (0, n_k) \}\) can only cover up to \((0, \max(n_1, n_2, \ldots, n_k))\), which leaves numbers greater than this maximum uncovered.
04

Conclusion of Non-Existence of Finite Subcover

Since no finite subcollection of \(\{ (0, n) \,|\, n \in \mathbb{N} \} \) can cover \([0, \infty)\), we conclude that \([0, \infty)\) does not have the Heine-Borel property. This demonstrates directly that an infinite subcover is necessary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Open Covers
An open cover of a set is a collection of open sets whose union contains the entire set we are interested in. To understand open covers better, think of having a blanket made of several small patches (open sets). These patches, when put together, entirely cover the area of interest (the set).
So when we say a set has an "open cover," we mean that we can use open intervals whose collective length spans and covers every part of the original set.
  • Each open set in an open cover may overlap, but together, they must cover every point in the set.
  • Open covers can be infinite, meaning you could potentially have an endless number of open intervals covering your set.
For example, if we look at the interval \( [0, \infty)\), an open cover could consist of intervals like \( (0, 1), (0, 2), (0, 3), \ldots\). Each \( (0, n)\) covers increasingly larger parts of the interval \( [0, \infty)\) as \( n\) gets larger.
Finite Subcover
A finite subcover refers to a smaller collection of open sets from an open cover that still covers the entire set but only uses a finite number of those open sets. Imagine picking only a few patches from your blanket, yet still covering the whole sofa to stay warm.
In terms of sets, if you can choose a finite number of intervals from your initial open cover that still blanket the entire set, then you have a finite subcover.
  • Having a finite subcover is key in determining if a set has the Heine-Borel property.
  • If any of the open cover cannot be reduced to a finite subcover that covers the set, then the set is not compact according to the Heine-Borel theorem.
For \( [0, \infty)\), you would attempt to find such a finite subset of intervals \( (0, n_1), (0, n_2), \ldots, (0, n_k)\), but typically you can't cover the infinite stretch completely with a finite number of intervals.
Unbounded Intervals
An unbounded interval is one that stretches out to infinity in at least one direction. It doesn鈥檛 have bounds on one end, meaning that it keeps going indefinitely. For example, \( [0, \infty)\) is an unbounded interval because it doesn't have an upper limit; it just keeps extending to the right without an end.
  • Unbounded intervals often do not fulfill the compactness condition required by the Heine-Borel theorem, which necessitates that intervals be both closed and bounded.
  • While open covers may apply to unbounded intervals, finding a finite subcover might not be possible because you cannot limit an infinite extent with a finite number of patches comfortably.
Recognizing whether an interval is bounded or unbounded can help determine whether it might have the Heine-Borel property, where compactness fails if either the bounded or closed condition is absent.

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Most popular questions from this chapter

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In many applications of open sets and closed sets we wish to work just inside some other set \(A\). It is convenient to have a language for this. A set \(E \subset A\) is said to be open relative to \(A\) if \(E=A \cap G\) for some set \(G \subset \mathbb{R}\) that is open. A set \(E \subset A\) is said to be closed relative to \(A\) if \(E=A \cap F\) for some set \(F \subset \mathbb{R}\) that is closed. Answer the following questions. (a) Let \(A=[0,1]\) describe, if possible, sets that are open relative to \(A\) but not open as subsets of \(\mathbb{R}\). (b) Let \(A=[0,1]\) describe, if possible, sets that are closed relative to \(A\) but not closed as subsets of \(\mathbb{R}\). (c) Let \(A=(0,1)\) describe, if possible, sets that are open relative to \(A\) but not open as subsets of \(\mathbb{R}\). (d) Let \(A=(0,1)\) describe, if possible, sets that are closed relative to \(A\) but not closed as subsets of \(\mathbb{R}\).
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