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

• I
• Math Amateur
In summary, in Theorem 3.28 of Apostol's "Mathematical Analysis" (Second Edition), it is proven that for any set A, the set of all n-balls A_m(x) obtained as x varies over A is a countable collection of open sets that covers A. This also holds true for uncountably infinite sets, as the chosen rationals in the proof form a countable subset of the set of all rationals.
Math Amateur
Gold Member
MHB
TL;DR Summary
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

Last edited:
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!

Last edited by a moderator:
Math Amateur and PeroK
Math Amateur said:
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.

member 587159

## What is the Lindelof Covering Theorem?

The Lindelof Covering Theorem is a mathematical theorem that states that every open cover of a compact set in a metric space has a countable subcover. In simpler terms, it means that any collection of open sets that completely cover a compact set can be reduced to a countable number of open sets that still cover the set.

## Who is Apostol and what is Theorem 3.28?

Apostol refers to Tom M. Apostol, a mathematician and professor at the California Institute of Technology. Theorem 3.28 is a theorem in his book "Mathematical Analysis" which states the Lindelof Covering Theorem.

## What is a compact set in a metric space?

A compact set in a metric space is a set that is closed and bounded. This means that the set contains all of its limit points and can be contained within a finite distance.

## Why is the Lindelof Covering Theorem important?

The Lindelof Covering Theorem is important because it provides a way to simplify complex open covers in a compact set. This can be useful in many areas of mathematics, such as topology and functional analysis.

## How is the Lindelof Covering Theorem used in real-world applications?

The Lindelof Covering Theorem has various applications in real-world situations, such as in physics, engineering, and economics. It can be used to prove the existence of solutions to certain equations, to analyze the behavior of systems, and to optimize processes.

Replies
2
Views
2K
Replies
2
Views
1K
Replies
4
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
1
Views
718
Replies
2
Views
1K
Replies
4
Views
2K
Replies
8
Views
2K
Replies
4
Views
2K