Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof)

In summary, the conversation is about understanding Proposition 4.1.1 on topology and continuity in Sohrab's book "Basic Real Analysis" (Second Edition). The main questions are about the interpretation of the proof and the process of constructing a countable subcollection. It is clarified that the set of all rational numbers is countable and any subset of rational numbers is also countable. The \rho_n or \rho_k in the proof are defined to be rational, making the subset \{ \rho_x \ : \ x \in O \} \subset \mathbb{Q} countable.
  • #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 \(\displaystyle \mathbb{R}\) and Continuity ... ...

I need help in order to fully understand the proof of Proposition 4.1.1...Proposition 4.1.1, some preliminary notes and its proof read as follows:
View attachment 9081
My questions are as follows:Question 1

In the above proof by Sohrab we read the following:

" ... ... Now the set \(\displaystyle \{ \rho_x \ : \ x \in O \} \subset \mathbb{Q}\) is countable ... ... "

But ... it seems to me that since the \(\displaystyle x\)'s are uncountable that the number of \(\displaystyle \rho_x\) is uncountable ... but that many (at times infinitely many ... ) have the same values since each is equal to a rational number and these are countable ...

... so in fact there are an uncountably infinite number of open balls \(\displaystyle B_{ \rho_x } (x)\) ... there are just a countable number of different values for the radii of the open balls ...

Is my interpretation correct ... ?

Question 2

In the above proof by Sohrab we read the following:

" ... ... If for each \(\displaystyle k \in \mathbb{N}\) we pick \(\displaystyle \lambda_k \in B_{ \rho_k } (x_k) \subset O_{ \lambda_k }\), then we have a countable subcollection \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\) which satisfies \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ... ..."Can someone please explain/demonstrate clearly (preferably in some detail) how the process described actually results in a countable subcollection where \(\displaystyle O = \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k }\) ...


... indeed, given that \(\displaystyle \{ O_{ \lambda_k } \}_{ k \in \mathbb{N} } \subset \{ O_\lambda \}_{ \lambda \in \Lambda }\)

... it looks as if \(\displaystyle \bigcup_{ k = 1 }^{ \infty } O_{ \lambda_k } \subset O\) ... ?

In addition to answers to the two questions, any explanations/clarifications of the overall strategy and tactics of the proof would be very gratefully received ...

Help will be much appreciated ...

Peter
 

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  • #2
The [tex]\rho_n[/tex] are defined to be rational. The set of all rational numbers is countable so this subset is countable.
 
  • #3
HallsofIvy said:
The [tex]\rho_n[/tex] are defined to be rational. The set of all rational numbers is countable so this subset is countable.
Thanks HallsofIvy ...

Yes, the \(\displaystyle \rho_n\) or \(\displaystyle \rho_k\) are constructed or defined to be rational ... but the \(\displaystyle \rho_x\) are also defined/constructed to be rational ... so my comments under question 1, I think, are correct ... do you agree ... am I interpreting the situation correctly ...

Can you comment on Question 2 ...?

Peter
 
  • #4
No, the set of all rational numbers is countable. Any set of rational numbers is at most countable. If a set of rational numbers, $\{r_x\}$, is indexed by "x" from an uncountable set that just means that there are a lot of duplicates- there are uncountably many different "x" such that "$r_x$" are the same.
 
  • #5
HallsofIvy said:
No, the set of all rational numbers is countable. Any set of rational numbers is at most countable. If a set of rational numbers, $\{r_x\}$, is indexed by "x" from an uncountable set that just means that there are a lot of duplicates- there are uncountably many different "x" such that "$r_x$" are the same.
Thanks HallsofIvy ...

Peter
 

Related to Compact Subsets of R .... Sohrab, Proposition 4.1.1 (Lindelof)

1. What is a compact subset of R?

A compact subset of R is a subset of the real numbers that is both closed and bounded. This means that every sequence in the subset has a convergent subsequence that also belongs to the subset.

2. How is compactness defined in this proposition?

In this proposition, compactness is defined as the property of a subset of R where every open cover has a finite subcover. This means that the subset can be covered by a finite number of open sets.

3. What is the significance of Proposition 4.1.1 (Lindelof) in mathematics?

This proposition is significant because it provides a characterization of compact subsets of R, which is a fundamental concept in topology. It also helps to establish a connection between compactness and other important properties, such as separability and metrizability.

4. Can you give an example of a compact subset of R?

One example of a compact subset of R is the closed interval [0,1]. This subset is bounded and closed, and any sequence within it has a convergent subsequence that also belongs to the subset.

5. What are the practical applications of understanding compact subsets of R?

Understanding compact subsets of R is important in various areas of mathematics, such as analysis, geometry, and topology. It also has applications in physics, engineering, and computer science, particularly in the fields of optimization and control theory.

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