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

Question

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?

Answer 1

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.

Answer 2

A sequence is a special case of an indexed family, namely one indexed over the natural numbers.

Answer 3

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.