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 finite direct products (or when all but finitely many of the factors are trivial). Definition Given such that or the (possibly infinite) Cartesian product of the topological spaces , indexed by , the box topology on is generated by the base The name box comes from the case of Rn, in which the basis sets look like boxes. The set endowed with the box topology is sometimes denoted by Properties Box topology on Rω: The box topology is completely regular The box topology is neither compact nor connected The box topology is not first countable (hence not metrizable) The box topology is not separable The box topology is paracompact (and hence normal and completely regular) if the continuum hypothesis is true Example — failure of continuity The following example is based on the Hilbert cube. Let Rω denote the countable cartesian product of R with itself, i.e. the set of all sequences in R. Equip R with the standard topology and Rω with the box topology. Define: So all the component functions are the identity and hence continuous, however we will show f is not continuous. To see this, consider the open set Suppose f were continuous. Then, since: there should exist such that But this would imply that which is false since for Thus f is not continuous even though all its component functions are. Example — failure of compactness Consider the countable product where for each i, with the discrete topology. The box topology on will also be the discrete topology.