I'm a computer science student currently studying universal algebra by reading the book Foundations of Algebraic Specification and Formal Software Development by Donald Sannella and Andrzej Tarlecki.
I'm having a hard time trying to understand the subtleties of indexed families. As far as I know, families are just maps between two sets: $I$, the index set; and $A$, the indexed set. That means that $|A|_{i_0}$, which is the element of $A$ indexed by $i_0$ could be basically anything, like a number, a set, a collection, etc. And that the particular case where $I$ is $\mathbb{N}$ it is called a sequence.
However, the definition of the product of an indexed family is:
$ \prod_{i\in I} A_i=\{f:I\to \bigcup_{i\in I} A_i: (\forall i_0\in I)(f(i_0)\in A_{i_0})\}$
Considering the definition of a family, then the product could be also defined as the set of all families $ (a_i)_{i \mathop \in I}$ with $|a|_{i_0} \in |A|_{i_0}$ for each $i_0 \in I$
But then, it is implied that the indexed set ($A$) must be a collection (a set of sets), because otherwise, $|A|_{i_0}$ could be an element like a number thus the union of $|A|_i$ would make no sense, since it only works for sets. So why does the definition of a family (at least the ones I've read on several math textbooks) does not require the indexed set to be a collection rather than just a set?
Also, would this mean that the elements of a sequence must be sets (containing just one or more elements)? I'm use to thinking about elements of a sequence, specially when studying convergence, as just real numbers.
And finally, I will give an example of what I understand so that you could tell me what I'm missing:
I will define a family of countries indexed by their currency name. So:
$I$, the index set would be $I = \{pound, dollar,euro\} $
$A$, the indexed set would be $A = \{\{Spain, Italy, France\},\{UK\},\{US, Canada\}\}$
Then, the family would be a mapping defined as: $(A_i)_{i \in I}= \{pound \rightarrow \{UK\},dollar \rightarrow \{US, Canada\}, euro \rightarrow \{Spain, Italy, France\} \} $
Finally, the product of the family $\prod_{i\in I} A_i$ will consist on a set of mappings (families) where each one of them has $I$ as index set, and maps each index to a set containing just one of the elements of the corresponding subset of $A$. So in that way, each of these families will map the indexes to one of the six combinations possibles by picking one element from each subset of $A$. Then the product of the family will be a set of exactly 6 families, somewhat similar to the Cartesian product of the subsets of $A$.
Best Answer
Families are not maps; unless that book has explicifly given the word family some other meaning, a family of things is simply a set of things. An indexed family of things, on the other hand, is technically a function from the index set to the unindexed set of those same things; the things themselves can be of any type(s). In practice, though, it’s often simpler to think of the indexing simply as a way of attaching labels to the members of the family.
As you’ve observed, the definition that you’ve been given of the product $\prod_{i\in I}A_i$ of the indexed family $\{A_i:i\in I\}$ applies only to indexed families of sets; that does not mean that you cannot have indexed families of other things, as in your example. However, your set
$$A=\big\{\{\text{Spain},\text{Italy},\text{France}\},\{\text{UK}\},\{\text{US},\text{Canada}\}\big\}$$
is not in itself an indexed family; it’s just a set of sets. It doesn’t become an indexed family until you actually index it. Taking $I=\{\text{dollar},\text{euro},\text{pound}\}$, you can index $A$ as $\{C_i:i\in I\}$, where
$$C_{\text{dollar}}=\{\text{US},\text{Canada}\}\,,$$ $$C_{\text{euro}}=\{\text{Spain},\text{Italy},\text{France}\}\,$$
and $$C_{\text{pound}}=\{\text{UK}\}\,.$$
Then $\prod_{i\in I}C_i$ is the set of all functions $f:I\to\bigcup_{i\in I}C_i$ such that $f(i)\in C_i$ for each $i\in I$. There are, as you say, six of them; one is
$$\big\{\langle\text{dollar},\text{US}\rangle,\langle\text{euro},\text{Italy}\rangle,\langle\text{pound},\text{UK}\rangle\big\}\,,$$
and the other five are similar. This is indeed similar to the Cartesian product of the sets $C_i$ for $i\in I$: for instance, this function corresponds to the ordered triple
$$\langle\text{US},\text{Italy},\text{UK}\rangle$$
in the product $C_{\text{dollar}}\times C_{\text{euro}}\times C_{\text{pound}}$.