Box Topology Name

Question

To put it simply, why is the box topology on an infinite product of topological spaces called the box topology? What is "boxy" about it?

Answer 1

Say we have a product $\prod_{a\in A}X_a$ of topological spaces $X_a$ for some index set $A$. Take open subsets $U_a\subseteq X_a$ for all $a\in A$. Then the "box" $\prod_{a\in A} U_a$ is open in the box topology (in fact, these sets form the standard basis for the box topology).

Think $X_a=\Bbb R$ with standard topology and $U_a$ being intervals, and the geometric analogy is complete.