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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Sohrab - 1 - Theorem 4.1.10 ... ...   PART 1 ... .png
Sohrab - 2 - Theorem 4.1.10 ... ...   PART 2 ... .png



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




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



Sohrab - Proposition 4.1.8 ... .png



Sohrab - Proposition 4.1.9 ... .png



Hope that helps ...

Peter
 

Olinguito

Well-known member
Apr 22, 2018
251
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.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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