Box topology

In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal 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, 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 than the product topology, although the two agree in the case of direct products (or when all but finitely many of the factors are trivial).

Given ${\displaystyle X}$ such that

or the (possibly infinite) Cartesian product of the topological spaces ${\displaystyle X_{i}}$, indexed by ${\displaystyle i\in I}$, the box topology on ${\displaystyle X}$ is generated by the base

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