Product topology on an arbitrary product

Definition

Let $(X_i)_{i \in I}$ be a family of topological spaces, indexed by any set $I$, and let $X = \prod_{i \in I} X_i$ be the Cartesian product, with projections $\pi_i \colon X \to X_i$. The product topology on $X$ is the topology generated by the subbasis $$\{ \pi_i^{-1}(U) : i \in I, \ U \text{ open in } X_i \}.$$

Remarks

A basis consists of the sets $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$. For two factors, this is the product topology of $X_1 \times X_2$. It is the coarsest topology on $X$ for which each projection $\pi_i$ is continuous. A sequence converges in $X$ if and only if each of its coordinates converges. The box topology is finer and lacks this property.

Explanations

L1 human

In an infinite product, a basic open set can restrict only finitely many coordinates. This choice makes the universal property and Tychonoff's theorem work.