# Understanding Indexed Families

I'm having a terrible time trying to understand what indexed families are. I read the wiki here http://en.wikipedia.org/wiki/Indexed_family but I found it so confusing.

Here's what I understood so far:

• A family is a set of subsets of a set. For example if $S=\{a,b,3,9\}$ then a family of $S$ could be $F = \{A1,A2,A3\}$ where $A1=\{a,3\}, A2=\{a,b,3\}, A3=\{b,9\}$, so $F$ could also possible be the union of $A1,A2,A3$

What I'm confused about and still don't understand:

• A family of numbers is a set of numbers or a set of subsets?
• An indexed family is a function that maps from an indexing set to ... what?
• Is the result of an indexed family is also a family? i.e. is it a set of subsets or a set of elements?

I hope if you can clarify with examples.

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The idea that a family is a set of subsets of some set is definitely wrong. – Christian Blatter Apr 19 '14 at 13:11

Indexing consists in associating, to each index $i$ in some set $I$, a unique element $a_i$ in some set $A$. This is exactly the definition of a function $a:I\to A$. The notation $a(i)$ and $a_i$ are therefore equivalent.
Historically, indexes have been used more frequently for functions $a:\mathbb{N}\to\mathbb{R}$. These are normally called sequences because we can write its values in a list: $a_1,a_2,a_3,...$.
Sometimes the index set, $I$, is needed to be larger, or to have other orderings, than the natural numbers $\mathbb{N}$. For example, in topology on often needs to consider a collection of open sets $U_i$ and, say, write down their union as $\bigcup_{i\in I} U_i$. The cardinality of $I$ becomes a way of expressing how many open sets are we considering.