I'm looking over some stuff from metric spaces and I came across the familiar theorem:(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]\left(X,d\right)[/tex] be a metric space and let [tex] \left\{ U_\alpha \right\}_{\alpha \in A} [/tex] be a family of open subsets of [tex]X[/tex]. Then the union of the family [tex] \left\{U_\alpha\right\}_{\alpha \in A }[/tex] is an open subset of [tex]X[/tex].

The proof is straightforward -- that's not my issue. My question is, why was this theorem stated using the idea of a "family of subsets" instead of a "set of subsets?" This same idea of "family of sets" pops up in the definition of a topology also, so I want to make sure I understand it.

I resorted back to Halmos'Naive Set Theoryand he says, "Observe that there is no loss of generality in considering families of sets instead of arbitrary collections of sets;...", so why don't they just say an arbitrary union of sets?

What am I missing here? What's the point of speaking of families?

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# Family vs. Set

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