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.
  • #1
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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:
View attachment 9097
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:
View attachment 9099
View attachment 9100
Hope that helps ...

Peter
 

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  • Sohrab - Proposition 4.1.8 ... .png
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  • #2
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.
 
  • #3
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
 

Related to Heine-Borel Theorem .... Sohrab, Theorem 4.1.10 .... ....

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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