# Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1

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## Definition

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an 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:

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

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 $i \in I$:

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

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 9$: Families

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.The definitions as given in these links are not strictly the entity as defined here -- they are variously countable or finite, indexed specifically by $\N$ or a subsetIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products