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For example, $A_i$ be the set of all numbers divisible by $i$, consider:

$\bigcup_{n \in \mathbb{N}} A_n$

Is this union a countable union because I am enumerating over a countable set (the natural numbers)?

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    $\begingroup$ Yes, “countable” here is purely a description of the indexing set over which the union is taken. Note that a countable union could equal an uncountable union. $\endgroup$ – MPW Nov 12 '17 at 21:22
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Yes. Countable union means the index set over which one is taking the union is countable. In this case it is $\mathbb{N}$, which is countable.

Incidentally, in this case, each of the $A_{i}$'s also happen to be countable, so what you have is a countable union of countable sets.

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