Definition:Indexing Set/Indexed Set

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 elements $i \in I$ under $x$.

The image of $x$, that is, $x \sqbrk I$ or $\Img x$, is called an indexed set.

That is, it is the set indexed by $I$.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations