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Equivalent Statements to Compactness ... Another Question ... Stromberg, Theorem 3.43 ... ...

Peter

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

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "


I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the \(\displaystyle V_k\) ...

... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...




Hope someone can help ...

Peter
 
Last edited:

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,679
At about the middle of the above proof by Stromberg we read the following:

" ... ... Otherwise enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ V_k \}_{ k = 1 }^{ \infty }\). ... ... "


I am wondering what are the \(\displaystyle V_k\) ... are they elements of \(\displaystyle \mathscr{V}\) (... that is, the \(\displaystyle U_B\)) ... or are they sets of some kind ... ... can someone please explain and elucidate the nature of the \(\displaystyle V_k\) ...

... indeed maybe the \(\displaystyle V_k\) are just equal to the \(\displaystyle U_B\) ... in that case why not enumerate \(\displaystyle \mathscr{V}\) as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) ...
I haven't read this proof carefully, but I am sure that you are correct: the elements of \(\displaystyle \mathscr{V}\) are exactly the sets \(\displaystyle U_B\), and it would be quite permissible to enumerate them as \(\displaystyle \{ U_{ B_k } \}_{ k = 1 }^{ \infty }\) (though that implies that you have enumerated the sets \(\displaystyle U_B\)). I think that Stromberg found it better to enumerate the sets in \(\displaystyle \mathscr{V}\) directly, rather than indirectly by enumerating the sets in $\mathscr{B}$. In that way, he avoids cumbersome double subscripts.