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Let $X$ be some nonempty set. In my analysis course, a family was described as a function $f : I \rightarrow X$ where $I$ is supposed to be an "index set" and we would write the family as $(x_i)_{i\in I}$ where $x_i \in X$. Can I describe every set as a family? What are the restrictions on the "index set", can every set be an "index set"?

I'm sorry if my questions don't make much sense to set theory scholars, I have only studied naive set theory so far (Still, I'd like a highly formalized answer if possible).

user159517
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  • Yes, an "index set" is just a set. If $X$ is a set I can always write $X = (f_x)_{x \in X}$, taking $f$ the identity. –  Mar 07 '15 at 15:11
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    The "index set" $I$ is a generalization of the traditional "numerical indexing" of successions : $a_i, i \in \mathbb N$. Trivially, we can have ; $f : X \to X$ and thus $f(x)=x$ for any $x \in X$. In this way $X$ is the index set itself... but I see no real benefit. – Mauro ALLEGRANZA Mar 07 '15 at 15:13
  • @N.H.: I'd balk at writing a set as a family in that way. See e.g. here for an example of how a conflation of sets with families could lead to real confusion. But one could write $1_X = (x){x \in X}$; and although there's little point in doing only that, one could generally use 'family' notation as a notation for functions, thus $f = (f(x)){x \in X}$, which is at least loosely comparable to set comprehension notation, as in e.g. $f(X) = {f(x): x \in X}$. But in most situations this would probably be awkward. – Calum Gilhooley Mar 07 '15 at 15:25
  • You're right it's not very the best notation. –  Mar 07 '15 at 15:32
  • @N.H.: It's a pity there isn't a more 'formal', subscript-free notation for functions, such as e.g. $f = [f(x): x \in X]$. But enough of this! I mustn't wander further off-topic. – Calum Gilhooley Mar 07 '15 at 15:47

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In general, given a set $X,$ an index set for $X$ is a set $I$ for which there is some surjection (i.e.: onto function) $f:I\to X.$ Such an $f$ is called an indexing (function) of $X$ (by $I$). Some texts make the further restriction that an indexing function should also be injective (i.e.: one-to-one), and so bijective (i.e.: a one-to-one correspondence). This seems to be what your text is describing as a family.

Regardless, we can always index a set by itself, letting, say $x_i:=i$ for all $i\in X$ (just the identity function on $X$!) so that the family is given by $(x_i)_{i\in X}.$

Now, this particular indexing is not of any particular utility, but it does show that we can always consider a set as a family (in a sense). Some proofs are made easier by working with families instead of sets, so that's good to keep in mind.

Cameron Buie
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