Equivalent Statements to COmpactness .... Stromberg, Theorem 3.43 .... ....

In summary, Karl R. Stromberg's book "An Introduction to Classical Real Analysis" discusses Equivalent Statements to Compactness. The theorem 3.43 states that for a compact subset A of a topological space X, there exists a point z in X such that ρ(z, x)<ϵ/2.
  • #1
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Equivalent Statements to Compactness ... Stromberg, Theorem 3.43 ... ...

I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.43 on pages 105-106 ... ... Theorem 3.43 and its proof read as follows:View attachment 9144
View attachment 9145

At about the middle of the above proof by Stromberg we read the following:

" ... ...Next select \(\displaystyle x \in A\) such that \(\displaystyle \rho (x, z) \lt \epsilon / 2\) [\(\displaystyle A\) is dense] ... ... "My question is as follows:

Can someone demonstrate rigorously how \(\displaystyle A\) is dense in \(\displaystyle X\) guarantees that we can select \(\displaystyle x \in A\) such that \(\displaystyle \rho (x, z) \lt \epsilon / 2\) ... ...
Stromberg defines dense in X as follows ... ...

A set \(\displaystyle A \subset X\) is dense in \(\displaystyle X\) if \(\displaystyle A^{ - } = X\).

Hope someone can help ...

Peter
========================================================================================It may help MHB readers to have access to Stromberg's terminology associated with topological spaces ... so I am providing access to the main definitions ... as follows:
View attachment 9146

Hope that helps ...

Peter
 

Attachments

  • Stromberg - 1 - Theorem 3.43 ... ... PART 1 ... .... .png
    Stromberg - 1 - Theorem 3.43 ... ... PART 1 ... .... .png
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  • Stromberg - 2 - Theorem 3.43 ... ... PART 2 ... .... ... .png
    Stromberg - 2 - Theorem 3.43 ... ... PART 2 ... .... ... .png
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  • Stromberg - Defn 3.11 ... Terminology for Topological Spaces ... .png
    Stromberg - Defn 3.11 ... Terminology for Topological Spaces ... .png
    24.6 KB · Views: 101
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  • #2
You have "A set A is dense in X if $D^-= X$. Doesn't Stromberg define "$D^-$ for a set A? Is it defined as the closure of A? I would think that would be $A^-$ rather than "D"! Where did the "D" come from?

(Actually the definition given later in what you copied is for "set D", not "set A".)

Your question
"
Can someone demonstrate rigorously how [FONT=MathJax_Math]A[/FONT] is dense in [FONT=MathJax_Math]X[/FONT] guarantees that we can select [FONT=MathJax_Math]x[FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math]A[/FONT][/FONT] such that [FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT]"[/FONT][FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2"[/FONT][/FONT] makes no sense without saying what "z" is. In a part you did not copy z is given as any point in X. The definition of "A is dense in X" is that the closure of A is X. And the closure or A is A union its limit points. That is, if "A is dense in X" then every point of A is a limit point of X. That, in turn, means that, given any $\epsilon> 0$, every point of X, in particular, z, has some point of X, x, such that $d(z, x)< \epsilon$. But if $\epsilon> 0$, so is $\epsilon/2$ so we can as well use $\epsilon/2$. [FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2?" Doesn[/FONT][/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT][/FONT]​
 
Last edited by a moderator:
  • #3
HallsofIvy said:
You have "A set A is dense in X if $D^-= X$. Doesn't Stromberg define "$D^-$ for a set A? Is it defined as the closure of A? I would think that would be $A^-$ rather than "D"! Where did the "D" come from?

(Actually the definition given later in what you copied is for "set D", not "set A".)

Your question
"
Can someone demonstrate rigorously how [FONT=MathJax_Math]A[/FONT] is dense in [FONT=MathJax_Math]X[/FONT] guarantees that we can select [FONT=MathJax_Math]x[FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math]A[/FONT][/FONT] such that [FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT]"[/FONT][FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2"[/FONT][/FONT]makes no sense without saying what "z" is. In a part you did not copy z is given as any point in X. The definition of "A is dense in X" is that the closure of A is X. And the closure or A is A union its limit points. That is, if "A is dense in X" then every point of A is a limit point of X. That, in turn, means that, given any $\epsilon> 0$, every point of X, in particular, z, has some point of X, x, such that $d(z, x)< \epsilon$. But if $\epsilon> 0$, so is $\epsilon/2$ so we can as well use $\epsilon/2$.

[FONT=MathJax_Math]ρ[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2?" Doesn[/FONT][/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]ϵ[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Main]2[/FONT][/FONT]
​
Thanks for the help, HallsofIvy ...

Sorry about the typo in the definition of "\(\displaystyle A\) is dense in \(\displaystyle X\)" ... ... I have corrected it in my post above ...

Regarding the definition of \(\displaystyle z\), I left that for the reader to get from the scanned text ... ...Now ... you write:

" ... ... the closure of \(\displaystyle A\) is \(\displaystyle A\) union its limit points. That is, if "\(\displaystyle A\) is dense in \(\displaystyle X\)" then every point of \(\displaystyle A\) is a limit point of \(\displaystyle X\). ... ... "I am having trouble seeing exactly why this is true ... ...Can you please explain how/why

\(\displaystyle A^- = A \cup \{ x \in X \ : \ x \text{ is a limit point of } A \} = X \)

\(\displaystyle \Longrightarrow\) every point of \(\displaystyle A\) is a limit point of \(\displaystyle X\) ... ...
Hope you can help further ...

Peter

 

FAQ: Equivalent Statements to COmpactness .... Stromberg, Theorem 3.43 .... ....

What is the definition of compactness in mathematics?

Compactness is a property of topological spaces that captures the idea of being "small" or "finite". A topological space is compact if every open cover of the space has a finite subcover. In other words, every open cover of the space can be reduced to a finite number of open sets that still cover the space.

How does Stromberg's Theorem 3.43 relate to compactness?

Stromberg's Theorem 3.43 states that a topological space is compact if and only if every infinite subset of the space has a limit point. This theorem provides an alternative characterization of compactness, which can be useful in certain situations.

Can you give an example of a compact space?

Yes, the closed interval [0,1] in the real numbers is a compact space. This can be proven using the Heine-Borel theorem, which states that a subset of Euclidean space is compact if and only if it is closed and bounded.

What are some equivalent statements to compactness?

Aside from Stromberg's Theorem 3.43, other equivalent statements to compactness include the Bolzano-Weierstrass theorem (every bounded sequence in Euclidean space has a convergent subsequence), the sequential compactness theorem (every infinite sequence in a compact space has a convergent subsequence), and the Tychonoff theorem (the product of any collection of compact spaces is compact).

How is compactness used in mathematical proofs?

Compactness is a powerful tool in mathematical proofs, particularly in analysis and topology. It allows for the reduction of infinite sets or spaces to finite ones, making them easier to work with. Compactness is also closely related to continuity and convergence, and can be used to prove important theorems such as the extreme value theorem and the Heine-Cantor theorem.

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