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I read an article and find a statement in the proof that I highlighted below.

enter image description here

I do not understand, why we can do relabelling?
And I do not know, how to do relabelling based on statement above?
Can I relabel, for example, $I_{1,0}=X_2$, $I_{1,1}=X_3$, $I_{1,2}=X_4$, and $I_{2,0}=X_5$?
Thanks for any explanation.

user136524
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  • The point is that you have an infinite sequence of sets $I_{1,0},I_{1,1}$, etc., indexed by pairs of integers. But what you really want is a sequence of sets $X_1,X_2,\dots$, indexed by natural numbers. This can be achieved simply by changing the names of the sets appropriately! – Kenta S Nov 15 '22 at 04:41
  • @KentaS How if I relabel ${0}$ and $I_{n,i}$ for $n=1,2,\dots$, $i=1,2,\dots,m-1$ as ${X_n}{n\in\mathbb{N}}$ by $X_1={0}$ and $X_n=I{n,i}$ for $n>1$, thus we have $[0,1]=\bigcup_{n\in\mathbb{N}}X_n$? Can I change the index $j$ in the picture above by index $n$? – user136524 Nov 15 '22 at 06:47
  • Then $I_{1,1}$ and $I_{1,2}$ will have the same label, but they may be different sets! – Kenta S Nov 15 '22 at 15:39
  • @KentaS oh, I see. So, I cannot change the index $j$ by index $n$, right? – user136524 Nov 17 '22 at 01:53

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