stylistic:style_transfer, structural:reformatting
Product topology

The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology.

More formally, given a sequence of spaces (X i) i ∈ N {\displaystyle (X_{i})_{i\in \mathbb {N} }}, the product space

is defined as the set of all sequences (x i) i ∈ N {\displaystyle (x_{i})_{i\in \mathbb {N} }} such that for each i, x i {\displaystyle x_{i}} is an element of X i {\displaystyle X_{i}}. The canonical projections are the maps p : X → X defined by the equation p i ((x j) j ∈ N) = x i {\displaystyle p_{i}((x_{j})_{j\in \mathbb {N} })=x_{i}}. Then 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 are continuous. The product topology is sometimes called the Tychonoff topology.