TITLE: Cartesian products of families in Halmos' book.
QUESTION [11 upvotes]: I'm studying some set theory from Halmos' book. He introduces the generalization of cartesian products by means of families. However, I can't understand what is going on. I get the first introduction "The notation..." to "... one-to-one correspondence". What I'm having trouble is with 

If $\{X_i\}$ is a family of sets $(i\in I)$, the Cartesian product of the family is, by definition, the set of all families $\{x_i\}$ with $x_i\in X_i$ for each $i$ in $I$.

Could you explain to me the motivation of this definition? I know families  are itself functions $f:I\to X$ such that to each $i$ there corresponds a subset of $X$, $x_i$. Instead of this we write them succintly as $\{x_i\}_{i\in I}$ to put emphasis on the range (indexed sets) of the function and the domain (indexing set) in question.
For example, in my case, the family is $f:I\to X$ with $f(i)=A_i$ with ${\rm dom} f=\{0,1,2,3\}$ and ${\rm ran} f =\left\{ {{A_0},{A_1},{A_2},{A_3}} \right\}$.
I'm thinking that we can talk about the cartesian product of sets as a set of tuples. However, I can't understand the definition for families of sets.
I leave the page in question:
$\hspace{1 cm} $

REPLY [12 votes]: I think this will help other readers that have this same question (Mendelson, Introduction to Topology):

Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$
as an ordered sequence such that $x_i\in X_i$. Given such a point, by setting $x(i)=x_i$ we obtain a function $x$ which associates to each integer $i$,$1\leq i \leq n$, the element $x(i)\in X_i$. Conversely, given a function $x$ which associates to each integer $i$,$1\leq i \leq n$, an element $x(i)\in X_i$ we obtain the point
$$(x(1),\dots,x(n))\in \prod_{i=1}^n X_i$$
It is easily seen that this correspondence between points of $\displaystyle\prod_{i=1}^n X_i$ and functions of the above type is one-one and onto, so that a point of $\displaystyle\prod_{i=1}^n X_i$ may also be defined as a function $x$ which associates to each integer $i$,$1\leq i \leq n$, a point  $x(i)\in X_i$. The advantage of this second approach is that it allows us to define the product of an arbitrary family of sets.
DEFINITION  Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product  of the sets $\{X_\alpha\}_{\alpha \in I}$, 
  written  $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$. 
Given a point $x\in \prod_{x\in I}X_\alpha$, one may refer to $x(\alpha)$ as the $\alpha$-th coordinate of $x$. However, unless the indexing set has been ordered in some fashion (as is the case with finite products in our earlier discussion), there is no first coordinate, second coordinate, and so on.