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It's my understanding that if I refer to a collection of sets of the form $A_i$, and refer to the intersection $$\bigcap_{i \in I} A_i$$ for some indexing set $I$, that I am directly implying that the $\{A_i\}$ are countable, which is even more obvious if I were to write $$\bigcap\limits_{i=1}^{\infty} A_i.$$ What If I were to specify that $I$ is an arbitrary (could be uncountable) indexing set? Would it then be valid to use the first formulation and refer to the set of $A_i$'s?

I've seen the term "arbitrary indexing set" used. I've also seen people write something that looked like the first form above without specifying that $I$ is arbitrary. I'm not sure which is standard or best practice.

John P.
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No, if you write $A_i, i \in I$ there is no implication that $I$ is countable. It can be any set and there is some implicit function that assigns to each $i \in I$ some set $A_i$.

I do say that if you write $$\bigcap_{i=1}^\infty A_i$$ instead, you're saying that $I=\mathbb{N}$, but this is confusing IMHO, it would then be clearer (in set context) to use $n,m$ as index variables to signal this. Or write $$\bigcap_{i \in \Bbb N} A_i$$

instead.

When you introduce the indexed family of sets in a proof, you also mention its index set, of course, and if you want it to be $\Bbb N$, name it $\Bbb N$. If your argument requires it to be arbitrary, just say "for some set $I$", and people will understand. Many authors just always use the same index set $I$ without mentioning it, or use $A$ with members $\alpha,\beta$; e.g. Munkres' text does this. It's a style thing too.

Henno Brandsma
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