Box topology

Definition

Let $(X_i)_{i \in I}$ be a family of topological spaces and let $X = \prod_{i \in I} X_i$ be the Cartesian product. The box topology on $X$ is the topology with basis the set of all products $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$.

Remarks

For finitely many factors, the box topology is the product topology. For infinite products, the box topology is finer than the product topology, and often strictly finer. On $\mathbb{R}^{\mathbb{N}}$ with the box topology, the map $t \mapsto (t, t, t, \ldots)$ from $\mathbb{R}$ is not continuous, although each coordinate is continuous. The preimage of $\prod_{n \ge 1} (-1/n, 1/n)$ is $\{0\}$, which is not open.

Explanations

L1 human

The box topology lets you restrict all coordinates at the same time. This looks natural, but it gives too many open sets, and simple maps into the product stop being continuous.