# Heine-Borel Theorem .... Sohrab, Theorem 4.1.10 .... ....

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In summary, the conversation focuses on understanding the proof of Theorem 4.1.10 in Chapter 4 of Houshang H. Sohrab's book, "Basic Real Analysis" (Second Edition). The proof involves finding a finite subcover of a compact set, and the question is raised about whether or not this subcover also covers a subset of the set. Olinguito clarifies that a finite subcover of the original cover is needed to prove the set is compact.
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 4: Topology of [FONT=MathJax_AMS]R[/FONT] and Continuity ... ...

I need help in order to fully understand the proof of Theorem 4.1.10 ... ... Theorem 4.1.10 and its proof read as follows:
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View attachment 9098
In the above proof by Sohrab we read the following:

" ... ...Since $$\displaystyle [a, b]$$ is compact (by Proposition 4.1.9) we can find a finite subcover $$\displaystyle \mathcal{O}'' \subset \mathcal{O}'$$ ... ..."My question is as follows:

If $$\displaystyle \mathcal{O}''$$ is a finite cover of $$\displaystyle [a, b]$$ then since $$\displaystyle K \subset [a, b]$$ surely $$\displaystyle \mathcal{O}'$$' is a finite cover of K also ... ... ?BUT ... Sohrab is concerned about whether or not $$\displaystyle \mathcal{O}' \in \mathcal{O}''$$ or not ... ...

Can someone please explain what is going on ...

Peter

========================================================================================The above post mentions Propositions 4.1.8 and 4.1.9 ... so I am providing text of the same ... as follows:
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Hope that helps ...

Peter

#### Attachments

• Sohrab - 1 - Theorem 4.1.10 ... ... PART 1 ... .png
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• Sohrab - 2 - Theorem 4.1.10 ... ... PART 2 ... .png
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• Sohrab - Proposition 4.1.8 ... .png
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• Sohrab - Proposition 4.1.9 ... .png
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Hi Peter.

$\cal O^{\prime\prime}$ is only a finite subcover of $\cal O^\prime$. In order to prove $K$ compact, we need to find a finite subcover of $\cal O$. That’s what’s going on.

Olinguito said:
Hi Peter.

$\cal O^{\prime\prime}$ is only a finite subcover of $\cal O^\prime$. In order to prove $K$ compact, we need to find a finite subcover of $\cal O$. That’s what’s going on.
Thanks Olinguito ...

That clarified the matter ...

Most grateful for you help ...

Peter

## 1. What is the Heine-Borel Theorem?

The Heine-Borel Theorem is a fundamental result in real analysis that states that a subset of Euclidean space is compact if and only if it is closed and bounded.

## 2. Who discovered the Heine-Borel Theorem?

The Heine-Borel Theorem was independently discovered by two mathematicians, Eduard Heine and Émile Borel, in the late 19th and early 20th century.

## 3. What is the significance of the Heine-Borel Theorem?

The Heine-Borel Theorem is significant because it provides a necessary and sufficient condition for a subset of Euclidean space to be compact. This allows for the simplification of many mathematical proofs and has applications in various fields such as analysis, topology, and physics.

## 4. How is the Heine-Borel Theorem related to the Bolzano-Weierstrass Theorem?

The Heine-Borel Theorem is closely related to the Bolzano-Weierstrass Theorem, which states that every bounded sequence in Euclidean space has a convergent subsequence. In fact, the Heine-Borel Theorem can be seen as a generalization of the Bolzano-Weierstrass Theorem to higher dimensions.

## 5. Can the Heine-Borel Theorem be extended to other metric spaces?

Yes, the Heine-Borel Theorem can be extended to other metric spaces, as long as they satisfy certain properties such as completeness and separability. This generalization is known as the Heine-Borel Covering Theorem.

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