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

[Math Amateur]

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

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

A set $$A \subset X$$ is dense in $$X$$ if $$A^{ - } = X$$.

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

 

[HallsofIvy]

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 is dense in [FONT=MathJax_Math]X guarantees that we can select [FONT=MathJax_Math]x[FONT=MathJax_Main]∈[FONT=MathJax_Math]A such that [FONT=MathJax_Math]ρ[FONT=MathJax_Main]([FONT=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2"[FONT=MathJax_Math]ρ[FONT=MathJax_Main]([FONT=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2" 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=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2?" Doesn[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2​

 

[Math Amateur]

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 is dense in [FONT=MathJax_Math]X guarantees that we can select [FONT=MathJax_Math]x[FONT=MathJax_Main]∈[FONT=MathJax_Math]A such that [FONT=MathJax_Math]ρ[FONT=MathJax_Main]([FONT=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2"[FONT=MathJax_Math]ρ[FONT=MathJax_Main]([FONT=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2"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=MathJax_Math]x[FONT=MathJax_Main],[FONT=MathJax_Math]z[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2?" Doesn[FONT=MathJax_Main])[FONT=MathJax_Main]<[FONT=MathJax_Math]ϵ[FONT=MathJax_Main]/[FONT=MathJax_Main]2
​

Thanks for the help, HallsofIvy ...

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

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

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

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

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

Peter