TITLE: Topology - The arbitrary union axiom
QUESTION [5 upvotes]: So, the common answer to why we need the concept of topology is that we need it to talk about things like limits of infinite sequences and continuity. But, when we define the axioms of topology, we have an axiom which says that an arbitrary(countable and uncountable) union of open sets exists(or is this not true?) and is open. But doesn't the notion of arbitrary union in itself hide an infinite sequence, namely the the sequence of partial unions of the open set? Isn't the union of infinitely many open sets equal to the "limit" of the partial unions? Is there an alternate interpretation of arbitrary unions which doesn't requires us to define a topology on the power set of the set of interest first?

REPLY [6 votes]: No, no sequence or limit process in involved. This is a purely set-theoretic concept.  If you have a collection of sets $\mathcal{F}$ in a universe of discourse $\Omega$, you define
$$\bigcup \mathcal{F} = \{x\in\Omega| \exists F\in \mathcal{F}\; {\rm with}\; x\in F\}.$$
It's just an existential quantifier. No order or structure is involved.