Is there a precise definition of "arbitrary union"? Is there a precise formulation of what "arbitrary union" means? For example, for a topology, we require that it be closed under arbitrary unions. Do we mean the union of any subcollection of the topology is also in the topology?
In general, do we define "arbitrary union" in terms of the union of a family? If so, how do we know this captures the idea of "arbitrary union" completely?
 A: Yes, if $\mathcal{T}$ is a collection of sets then it is closed under "arbitrary unions" if
$$\forall \mathcal{T}' \subseteq \mathcal{T}: \bigcup \mathcal{T}' \in \mathcal{T}$$
so in words: the union of any subfamily of the family is also in the family.
Note that this includes the finite unions: if $O_1, O_2 \in \mathcal{T}$ we can take $\mathcal{T}'=\{O_1,O_2\}\subseteq \mathcal{T}$ and then $\bigcup \mathcal{T}' = O_1 \cup O_2 \in \mathcal{T}$ e.g. And likewise for countable unions: if $O_n, n \in \Bbb N$ are in $\mathcal{T}$ , take $\mathcal{T}'=\{O_n\mid n \in \Bbb N\}$ and then $\bigcup_n O_n = \bigcup \mathcal{T'} \in \mathcal{T}$ etc.
Often we just write an arbitrary union as $\bigcup_{i \in I} O_i$ where $i \in I$, $I$ is some index set, and all $O_i \in \mathcal{T}$. Then we leave unspecified whether $I$ is finite, countable or whatever.
A: "Arbitrary union" essentially means that there are no limitations. We might specify in other cases that something is true for only finitely or countably many iterations of that operation, particularly in topology for instance, but arbitrary - as in "arbitrarily many" - means it holds no matter how many: finite, countable, uncountable...

Do we mean the union of any subcollection of the topology is also in the topology?

So a direct answer: yes.
A: Consider the union: ${\cup A_i}.$ Here the index $i \in I.$ For arbitrary union, the set $I$ could be anything; not necessarily a finite or countable one. The set $I$ could be an uncountable set like $[0, 1].$
