Definition:Indexing Set/Function

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Definition

Let $I$ and $S$ be sets.

Let $x: I \to S$ be a mapping.

Let $x_i$ denote the image of an element $i \in I$ of the domain $I$ of $x$.

Let $\family {x_i}_{i \mathop \in I}$ denote the set of the images of all the element $i \in I$ under $x$.


When used in this context, the mapping $x$ is referred to as an indexing function for $S$.


Notation

The family of elements $x$ of $S$ indexed by $I$ is often seen with one of the following notations:

$\family {x_i}_{i \mathop \in I}$
$\paren {x_i}_{i \mathop \in I}$
$\set {x_i}_{i \mathop \in I}$


There is little consistency in the literature, but $\paren {x_i}_{i \mathop \in I}$ is perhaps most common.

The preferred notation on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $\family {x_i}_{i \mathop \in I}$.

The subscripted $i \in I$ is often left out, if it is obvious in the particular context.


Note the use of $x_i$ to denote the image of the index $i$ under the indexing function $x$.

As $x$ is actually a mapping, one would expect the conventional notation $\map x i$.

However, this is generally not used, and $x_i$ is used instead.


Also defined as

Some treatments make the added stipulation that $x$ needs to be surjective for this definition to be valid. This presupposes the condition that the whole of $S$ is thus indexed.

However, the definition as given on $\mathsf{Pr} \infty \mathsf{fWiki}$ does not impose that condition. If such a condition is required for a specific construct, then surjectivity will be imposed as appropriate.


Also see

If the the indexing set $I$ is finite or countable (and in particular if $I \subseteq \N$), then the family $\family {x_i}_{i \mathop \in I}$ is often referred to as a sequence.


Note on Terminology

It is a common approach to blur the distinction between an indexing function $x: I \to S$ and the indexed family $\family {x_i}_{i \mathop \in I}$ itself, and refer to the mapping as the indexed family.

This approach is in accordance with the definition of a mapping as a relation defined as an as ordered triple $\tuple {I, S, x}$, where the mapping is understood as being defined to include its domain and codomain.


However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the approach is taken to separate the concepts carefully such that an indexed family is defined as:

the set of terms of the indexed set

together with:

the indexing function itself

denoting the combination as $\family {x_i}_{i \mathop \in I}$.


The various approaches in the literature can be exemplified as follows.

There are occasions when the range of a function is deemed to be more important than the function itself. When that is the case, both the terminology and the notation undergo radical alterations. Suppose, for instance, that $x$ is a function from a set $I$ to a set $X$. ... An element of the domain $I$ is called an index, $I$ is called the index set, the range of the function is called an indexed set, the function itself is called a family, and the value of the function $x$ at an index $i$, called a term of the family, is denoted by $x_i$. (This terminology is not absolutely established, but it is one of the standard choices among related slight variants...) An unacceptable but generally accepted way of communicating the notation and indicating the emphasis is to speak of a family $\set {x_i}$ in $X$, or of a family $\set {x_i}$ of whatever the elements of $X$ may be; when necessary, the index set $I$ is indicated by some such parenthetical expression as $\paren {i \in I}$. Thus, for instance, the phrase "a family $\set {A_i}$ of subsets of $X$" is usually understood to refer to a function $A$, from some set $I$ of indices, into $\powerset X$.
1960: Paul R. Halmos: Naive Set Theory: $\S 9$: Families


Occasionally, the special notation for sequences is also employed for functions that are not sequences. If $f$ is a function from $A$ into $E$, some letter or symbol is chosen, say "$x$", and $\map f \alpha$ is denoted by $x_\alpha$ and $f$ itself by $\paren {x_\alpha}_{\alpha \mathop \in A}$. When this notation is used, the domain $A$ of $f$ is called the set of indices of $\paren {x_\alpha}_{\alpha \mathop \in A}$, and $\paren {x_\alpha}_{\alpha \mathop \in A}$ is called a family of elements of $E$ indexed by $A$ instead of a function from $A$ into $E$.
1965: Seth Warner: Modern Algebra: $\S 18$: The Natural Numbers


Let $I$ and $E$ be sets and let $f: I \to E$ be a mapping, described by $i \mapsto \map f i$ for each $i \in I$. We often find it convenient to write $x_i$ instead of $\map f i$ and write the mapping as $\paren {x_i}_{i \mathop \in I}$ which we shall call a family of elements of $E$ indexed by $I$. By abuse of language we refer to the $x_i$ as the elements of the family.
...
As we have already mentioned, many authors identify a mapping with its graph, thereby identifying the family $\paren {x_i}_{i \mathop \in I}$ with the set $\set {\tuple {i, x_i}; i \in I}$. In the case where the elements of the family are all distinct, some authors go even further and identify the mapping $\paren {x_i}_{i \mathop \in I}$ with its image $\set {x_i; i \in I}$.
1975: T.S. Blyth: Set Theory and Abstract Algebra: $\S 6$. Indexed families; partitions; equivalence relations


Some authors are specific about the types of objects to which this construction is applied:

Let $\mathcal A$ be a nonempty collection of sets. An indexing function for $\mathcal A$ is a surjective function $f$ from some set $J$, called the index set, to $\mathcal A$. The collection $\mathcal A$, together with the indexing function $f$, is called an indexed family of sets. Given $\alpha \in J$, we shall denote the set $\map f \alpha$ by the symbol $\mathcal A_\alpha$. And we shall denote the indexed family itself by the symbol $\set {\mathcal A_\alpha}_{\alpha \mathop \in J}$, which is read as "the family of all $\mathcal A_\alpha$, as $\alpha$ ranges over $J$."
2000: James R. Munkres: Topology (2nd ed.): $\S 5$: Cartesian Products


Sources