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Suppose I have a infinite set $S$, and would like to label all the points in $S$ as $x_i$. Then for each $x_i$, I would like to take a set that contains $x_i$, called $U_i$ and take the union of all these $U_i$s.

Question: Is the following way of writing it correct and clear? Is it implied that $I$ is just the set that gives me a bijection with each $x_i \in S$?

Let $S = \{ x_i : i \in I \}$ where $I$ is an indexing set. For each $x_i$, let $U_i$ be a set such that $x_i \in U_i$. This is the union: $\bigcup\limits_{i \in I} U_i$

Snowball
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1 Answers1

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This might be technically correct, but the set $I$ is superfluous. $S$ itself can be used as its own indexing set (or, if you prefer, there is no need for indexing at all). You can simply write $$ \bigcup_{x\in S} U_x $$ to mean what you want. (It doesn't even matter if $S$ is infinite or finite or even empty.)

Greg Martin
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  • Yes that's much better. What I wrote did look odd, since the $I$ in my notation was sort of implied, which I had never seen before. Thank you! – Snowball Sep 25 '20 at 02:25