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- I need help in order to fully understand Tom M. Apostol's proof of the Lindelof Covering Theorem ...

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:

In the above proof by Apostol we read the following:

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

My question is as follows:

What happens when ##A## is an uncountably infinite set ... how does the set of all ##n##-balls ##A_{ m(x) }## remain as a countable collection of open sets which covers ##A## ... when ##x## ranges over an uncountable set ... ...? ... ...My thoughts are as follows: ... ... the sets ##A_{ m(x) }## must be used/repeated 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:

Hope that helps ...

Peter

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:

In the above proof by Apostol we read the following:

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

My question is as follows:

What happens when ##A## is an uncountably infinite set ... how does the set of all ##n##-balls ##A_{ m(x) }## remain as a countable collection of open sets which covers ##A## ... when ##x## ranges over an uncountable set ... ...? ... ...My thoughts are as follows: ... ... the sets ##A_{ m(x) }## must be used/repeated 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:

Hope that helps ...

Peter

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