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

#### Peter

##### Well-known member
MHB Site Helper
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:  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

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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: Hope that helps ...

Peter

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#### HallsofIvy

##### Well-known member
MHB Math Helper
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".)

"
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]

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#### Peter

##### Well-known member
MHB Site Helper
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".)

"
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 ... ...

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

$$\displaystyle \Longrightarrow$$ every point of $$\displaystyle A$$ is a limit point of $$\displaystyle X$$ ... ...

Hope you can help further ...

Peter