Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance.

There is also the direct sum – in some areas this is used interchangeably, in others it is a different concept.

In a similar manner, we can talk about the product of more than two objects, e.g. ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} \times \mathbb {R} }$. We can even talk about product of infinitely many objects, e.g. ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} \times \dotsb }$.

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