Definition:Indexing Set/Indexed Set

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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 $I$.