The box topology is the topology on the cartesian product of sets generated by the cartesian product of open subsets in each component set. Use this tag when your question involves the box topology.
The Cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is box topology, in which a base is given by Cartesian products of open sets in the component spaces. Another possibility is product topology, in which a base is given by Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.
While box topology has a somewhat more intuitive definition than product topology, box topology satisfies fewer desirable properties. In particular, if all the component spaces are compact/connected, the box topology on their Cartesian product will not necessarily be compact/connected, but the product topology on their Cartesian product will always be compact/connected. In general, box topology is finer than product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial).
