Definition:Indexing Set

Definition

Let $I$ and $S$ be sets.

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

Index

An element of the domain $I$ of $x$ is called an index.

Indexed Set

An element of the image of $x$ is called an indexed set.

Family

The mapping $x$ itself is called a family.

Term

The value of $x$ at an index $i$ is called a term of the family, and is denoted $x_i$.

Notation

The family $x$ is often seen with one of the following notations:

$\left \langle {x_i} \right \rangle_{i \in I}$

$\left({x_i}\right)_{i \in I}$

$\left\{{x_i}\right\}_{i \in I}$

There is little consistency in the literature.

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

The domain $I$ of $x$ is called the indexing set of $\left \langle {x_i} \right \rangle_{i \in I}$.

Thus the phrase:

a family $\left \langle {x_i} \right \rangle$ of subsets of $S$

is taken to mean:

a mapping $x$ from some indexing set $I$ into the power set $\mathcal P \left({S}\right)$ of $S$.

Alternative terms

Some authors use the term index set for indexing set, while others uses set of indices.

Also see

Compare the definition of a sequence, where the indexing set used is the set of natural numbers $\N$, or a subset of $\N$.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 9$: Families
• Seth Warner: Modern Algebra (1965): $\S 18$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.2$