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Concept# Box topology

Summary

In topology, the cartesian product of topological spaces can be given several different topologies. One of the more natural 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).
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
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
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.

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Box topology

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