Metric Spaces: Theorem of Open Unions & Families of Sets

[union68]

I'm looking over some stuff from metric spaces and I came across the familiar theorem:

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

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 Theory and 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?

 

[rasmhop]

They use the term family when they want to emphasize that they are all subsets of some set ($U_a \subseteq X$). Formally there is no difference, but you rarely speak about a family of integers because we don't think of 5 a set (even though according to most definitions it is). I have also heard it used simply to introduce some variety in the sentence structure (just as collection is used).

According to some definitions families of sets are allowed to be proper classes instead of sets. So for instance the class of all sets can be called the family of all sets, but not the set of all sets as that would produce a contradiction. That's not relevant in this case though as $P(X)$ is a set if $X$ is a set.

EDIT: Also when we index a collection of sets by an indexing sets we call it an indexed family of sets and sometimes we drop the term indexed (as in your case).

 

[bpet]

What's the point of speaking of families?

It's a matter of convenience that makes the statements easier to follow - especially for theorems where the structure or size of the index set matters (e.g. ordered, countable etc).