Definition:Cartesian Product/Countable

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:
  • Definition:External Direct Product/General Definition
  • Definition:Internal Direct Product/General Definition
  • Definition:Group Direct Product/General Definition
  • Definition:Internal Group Direct Product/General Definition

• Results about Cartesian products can be found here.

Source of Name

This entry was named for René 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 $15.5$