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

[Math Amateur]

TL;DR

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

 

[member 587159]

The balls $A_{m(x)}$ form a subset of $G$. G is countable, and thus since any subset of $G$ is countable, the result follows.

There is nothing in this proof that says that A isn't uncountable. The proof also works in this case.

Ps: Apostol is a great book. Good choice!

 

[PeroK]

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?

Yes. A simpler way to look at it is as follows. Suppose for every real number $x$ we choose a rational $r(x)$. Effectively this is a mapping from $\mathbb{R}$ to $\mathbb{Q}$.

The range of this function, i.e. the set of all rationals we chose, is clearly a subset of $\mathbb{Q}$, hence countable.

Corollary (your thought): there exists at least one rational $r$ that was chosen for an uncountably infinitely many $x$'s.