Topological product

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".

Given X such that

is the Cartesian product of the topological spaces X_i, indexed by ${\displaystyle i\in I}$, and the canonical projections p_i : X → X_i, the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections p_i are continuous. The product topology is sometimes called the Tychonoff topology.

The open sets in the product topology are unions (finite or infinite) of sets of the form ${\displaystyle \prod _{i\in I}U_{i}}$, where each U_i is open in X_i and U_i ≠ X_i for only finitely many i. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the X_i gives a basis for the product ${\displaystyle \prod _{i\in I}X_{i}}$.

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