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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:





Stromberg - 1 - Theorem 3.43 ... ... PART 1 ... .... .png
Stromberg - 2 - Theorem 3.43 ... ... PART 2 ... .... ... .png




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:



Stromberg -  Defn 3.11  ... Terminology for Topological Spaces ... .png




Hope that helps ...

Peter
 
Last edited:

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
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:

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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