# Definition:Cartesian Product/Countable

< Definition:Cartesian Product(Redirected from Definition:Countable Cartesian Product)

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## Definition

Let $\sequence {S_n}_{n \mathop \in \N}$ be an infinite sequence of sets.

The **cartesian product** of $\sequence {S_n}$ is defined as:

- $\ds \prod_{k \mathop = 1}^\infty S_k = \set {\tuple {x_1, x_2, \ldots, x_n, \ldots}: \forall k \in \N: x_k \in S_k}$

It defines the concept:

- $S_1 \times S_2 \times \cdots \times S_n \times \cdots$

Thus $\ds \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\tuple {x_1, x_2, \ldots, x_n, \ldots}$ with $x_k \in S_k$.

## Also see

**Generalized Cartesian products**of algebraic structures:

- Results about
**Cartesian products**can be found here.

## Source of Name

This entry was named for RenĂ© Descartes.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 11$: Numbers - 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

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- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.15$: Sequences: Definition $15.5$