# 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 $\displaystyle \prod_{i \mathop \in I} S_i$ 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 $i \in I$:

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

## Sources

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

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products