# What is the difference between an indexed family and a sequence?

For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J

For sequence wikipedia states: Formally, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.

Both can have repeated elements and their order matters. Is there any difference between this two?

• Please cite (some of) the definitions, and tell us where you get confused. – vonbrand Dec 30 '15 at 0:58

A sequence in a set $X$ is, intuitively, a list $x_1, x_2, x_3, \ldots$ of elements of $X$ (there is an $x_i$ for every positive integer $i$). Formally, this sequence is a function $f : \{1, 2, \ldots\} \to X$ given by $f(i) = x_i$.
An indexed family in $X$ is a collection of elements $x_i \in X$, indexed by $i \in I$, where $I$ can be any set. Formally, this indexed family is a function $f : I \to X$ given by $f(i) = x_i$.
Here you can see that a sequence is a special type of indexed family: one where the indexing set $I$ is the set of positive integers. Of course, with sequences you have the additional notion that the terms are in some sort of order (e.g., $x_1$ is "before" $x_2$). This notion isn't present with general indexed families, unless the indexing set has some sort of order relation defined on it.
You can index something by the real numbers for example. So for every real number $\alpha$ you might have a value $x_{\alpha}$. But since you cannot enumerate the real numbers you cannot represent that as a sequence. So it's just an indexed set, something more general than a sequence.