Definition:Cartesian Product/Cartesian Space/Family of Sets

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Definition

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

Let $\ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

Let $S$ be a set such that:

$\forall i \in I: S_i = S$


Definition 1

The Cartesian space of $S$ indexed by $I$ is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$:

$S_I := \ds \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$


Definition 2

The Cartesian space of $S$ indexed by $I$ is defined and denoted as:

$\ds S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$


Also known as

It is reported that some sources give this as the Cartesian $I$-space of $S$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ it is generally referred to as an indexed Cartesian space.


Examples

Singleton Index

Let $X$ be a set.

Then the indexed Cartesian space $X^{\set 1}$ is defined as:

$X^{\set 1} = \set 1 \times X$


Sources


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