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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

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