Box topology is a topology that can be applied to products of topological spaces, especially in the context of infinite product spaces. It is defined on the Cartesian product of a collection of topological spaces, and it has some distinct properties compared to another common topology used on product spaces, known as the product topology.
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The box topology on has a basis of a topology consisting of all Cartesian products with every an open set in ; infinitely many coordinates may be restricted. It contains the product topology and agrees with it for finite .[ref]
18.901, Fall 2024 lecture notes. Lecture 12, Exercise 12.1.2(1)-(2), printed page 43: box basis, comparison with the product topology, and finite-factor equality
For copies of the usual real line, the box topology is strictly finer when . The box is not an open set in the product topology: a basic cylinder set containing zero leaves some coordinate unrestricted, so it contains . This is a construction for every finite coordinate set, not evidence from testing a finite prefix.