Is a family a one-to-one function?

Question

I haven't seen this being explicitly stated anywhere. I'm reading Halmos' book on Naive Set Theory, and it defines a family as a function from an indexing set to an indexed set. It never really said that the function is one-to-one. Since, by intuition it seems that it is not practical for one element in the indexed set to be assigned two different indices. So as the title suggests, is it obviously implied that a family is a one-to-one function? Thanks in advance.

Answer 1

The function need not be one-to-one.

It is possible (and practical) for one element of the indexed set to receive two different indices. It just means that there are repeated elements in the family (and it also means that we allow repeated elements in a family).

A natural question is when do we need such tools? There are cases where we want to discuss more than just the elements of the set. An example that illustrates this is a sequence. The sequence $a_n = (-1)^n$ is a family indexed by $\mathbb{N}$. This family contains more information than the set $\{-1,1\}$. It is describing a movement between $-1$ and $1$ in an alternating manner.

Answer 2

No, it is allowed that the index function is non-injective. In some cases it is even extremely useful to work with non-injective functions. Consider for example products of sets $P=\prod_{i=1}^nX_i$. Here $\{X_i\}$ is an indexed family of sets. The index set is $\{1,\ldots,n\}$ and the index function associates the set $X_i$ to the index $i$. If we require that index functions have to be injective, then all factors of $P$ must be distinct and we would never get objects like $\mathbb R^n$ having identical factors $X_i= \mathbb R$.