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.[1] 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).

DefinitionEdit

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.

PropertiesEdit

Box topology on Rω:[2]

Example - failure of continuityEdit

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 compactnessEdit

Consider the countable product   where for each i,   with the discrete topology. The box topology on   will also be the discrete topology. Since discrete spaces are compact if and only if they are finite, we Immediately see that   is not compact, even though its component spaces are.

  is not sequentially compact either: consider the sequence   given by

 

Since no two points in the sequence are the same, the sequence has no limit point, and therefore   is not sequentially compact.

Convergence in the box topologyEdit

Topologies are often best understood by describing how sequences converge. In general, a Cartesian product of a space   with itself over an indexing set   is precisely the space of functions from   to  , denoted  . The product topology yields the topology of pointwise convergence; sequences of functions converge if and only if they converge at every point of  .

Because the box topology is finer than the product topology, convergence of a sequence in the box topology is a more stringent condition. Assuming   is Hausdorff, a sequence   of functions in   converges in the box topology to a function   if and only if it converges pointwise to   and there is a finite subset   and there is an   such that for all   the sequence   in   is constant for all  . In other words, the sequence   is eventually constant for nearly all   and in a uniform way.[3]

Comparison with product topologyEdit

The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : YXi into the component spaces: the product map f: YX defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples—many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology.

See alsoEdit

NotesEdit

  1. ^ Willard, 8.2 pp. 52–53,
  2. ^ Steen, Seebach, 109. pp. 128–129.
  3. ^ Scott, Brian M. "Difference between the behavior of a sequence and a function in product and box topology on same set". math.stackexchange.com.

ReferencesEdit

External linksEdit