Editing Box topology (section)

In [[topology]], the [[cartesian product]] of [[topological space]]s can be given several different topologies. One of the more [[Natural_topology|natural]] choices is the '''box topology''', where a [[Base (topology)|base]] is given by the Cartesian products of open sets in the component spaces.<ref>Willard, 8.2 pp. 52&ndash;53,</ref> Another possibility is the [[product topology]], where a base is also given by the Cartesian products of open sets in the component spaces, but only finitely many of which can be unequal to the entire component space.

While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are [[compact space|compact]], the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is [[finer topology|finer]] than the product topology, although the two agree in the case of [[wiktionary:finite|finite]] direct products (or when all but finitely many of the factors are [[trivial topology|trivial]]).