# Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....

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In summary, Peter is seeking help in understanding Theorem 3.28 (Lindelof Covering Theorem) from Tom M Apostol's book "Mathematical Analysis" (Second Edition). He is specifically focused on Chapter 3: Elements of Point Set Topology and is questioning how the set of n-balls A_{m(x)} remains a countable collection of open sets that covers an uncountably infinite set A when x ranges over an uncountable set. Opalg explains that there will be uncountably many points in A, but only countably many elements in A_k, so there will be multiple points x in A that give rise to the same element in A_k. Opalg also expresses
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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 3: Elements of Point Set Topology ... ...

I need help in order to fully understand Theorem 3.28 (Lindelof Covering Theorem ... ) .Theorem 3.28 (including its proof) reads as follows:View attachment 9082
View attachment 9083In the above proof by Apostol we read the following:

" ... ... The set of all $$\displaystyle n$$-balls $$\displaystyle A_{ m(x) }$$ obtained as $$\displaystyle x$$ varies over all elements of $$\displaystyle A$$ is a countable collection of open sets which covers $$\displaystyle A$$ ... ..."
My question is as follows:

What happens when $$\displaystyle A$$ is an uncountably infinite set ... how does the set of all $$\displaystyle n$$-balls $$\displaystyle A_{ m(x) }$$ remain as a countable collection of open sets which covers $$\displaystyle A$$ ... when $$\displaystyle x$$ ranges over an uncountable set ... ...?My thoughts are as follows: ... ... ... the sets $$\displaystyle A_{ m(x) }$$ must be used many times ... indeed in many cases infinitely many times ... is that correct?

Help will be much appreciated ...

Peter=====================================================================================The above post refers to Theorem 3.27 ... so I am providing text of the same ... as follows:View attachment 9084
View attachment 9085
Hope that helps ...

Peter

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• Apostol - 1- Theorem 3.28 ... PART 1 ... .png
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• Apostol - 2 - Theorem 3.28 ... PART 2 ... .png
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• Apostol - 1- Theorem 3.27 ... PART 1 ... .png
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• Apostol - 2- Theorem 3.27 ... PART 2 ... .png
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Peter said:
In the above proof by Apostol we read the following:

" ... ... The set of all $$\displaystyle n$$-balls $$\displaystyle A_{ m(x) }$$ obtained as $$\displaystyle x$$ varies over all elements of $$\displaystyle A$$ is a countable collection of open sets which covers $$\displaystyle A$$ ... ..."

My question is as follows:

What happens when $$\displaystyle A$$ is an uncountably infinite set ... how does the set of all $$\displaystyle n$$-balls $$\displaystyle A_{ m(x) }$$ remain as a countable collection of open sets which covers $$\displaystyle A$$ ... when $$\displaystyle x$$ ranges over an uncountable set ... ...?

My thoughts are as follows: ... ... ... the sets $$\displaystyle A_{ m(x) }$$ must be used many times ... indeed in many cases infinitely many times ... is that correct?
That is correct. There are uncountably many points in $A$, but there are only countably many elements $A_k$ in $G$. So (in general) there will be uncountably many different points $x\in A$ giving rise to the same element $A_k = A_{m(x)}\in G$.

I very much prefer Apostol's proof of the Lindelöf covering theorem to that of Sohrab which you quoted in https://mathhelpboards.com/analysis-50/compact-subsets-r-sohrab-proposition-4-1-1-lindelof-26249.html#post115993. I found Sohrab's proof almost impenetrable, but Apostol presents essentially the same argument in a much more transparent way.

Opalg said:
That is correct. There are uncountably many points in $A$, but there are only countably many elements $A_k$ in $G$. So (in general) there will be uncountably many different points $x\in A$ giving rise to the same element $A_k = A_{m(x)}\in G$.

I very much prefer Apostol's proof of the Lindelöf covering theorem to that of Sohrab which you quoted in https://mathhelpboards.com/analysis-50/compact-subsets-r-sohrab-proposition-4-1-1-lindelof-26249.html#post115993. I found Sohrab's proof almost impenetrable, but Apostol presents essentially the same argument in a much more transparent way.
Thanks for the help Opalg ...

The fact that you found Sohrab's proof almost impenetrable was such a relief to me ... since I found it. Incredibly difficult/impossible ... but I did follow Apostol ...

Thanks again ...

Peter

## 1. What is the Lindelof Covering Theorem?

The Lindelof Covering Theorem, also known as the Lindelof's lemma, is a fundamental result in topology that states that every open cover of a metric space has a countable subcover. In other words, if a metric space can be covered by a collection of open sets, then it can also be covered by a countable number of open sets.

## 2. Who is the mathematician behind the Lindelof Covering Theorem?

The Lindelof Covering Theorem was named after Finnish mathematician Ernst Leonard Lindelof, who first proved the theorem in 1907. However, the concept of a countable subcover was first introduced by French mathematician Émile Borel in 1895.

## 3. How is the Lindelof Covering Theorem applied in mathematics?

The Lindelof Covering Theorem is a powerful tool in topology, as it allows mathematicians to prove the existence of certain objects or properties. For example, it can be used to prove the existence of a continuous function between two topological spaces, or to show that a space is compact.

## 4. What is Theorem 3.28 in Apostol's "Mathematical Analysis" book?

Theorem 3.28 in Apostol's "Mathematical Analysis" book is a generalization of the Lindelof Covering Theorem for metric spaces. It states that if a metric space can be covered by a collection of open sets with a certain property, then it can also be covered by a countable number of open sets with the same property.

## 5. Can the Lindelof Covering Theorem be applied to non-metric spaces?

Yes, the Lindelof Covering Theorem can be extended to non-metric spaces, such as topological spaces. In this case, the theorem states that every open cover of a topological space has a countable subcover. This is a more general version of the theorem and is often used in topological proofs.

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