# 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).

## 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 **R**^{n}, the basis sets look like boxes or unions thereof.

## Properties

Box topology on **R**^{ω}:[2]

- 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 at continuity

The following example is based on the Hilbert cube. Let **R**^{ω} 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 **R**^{ω} 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 at compactness

Consider the countable product
where for each *i*,
with the discrete topology. The box topology on
will also be the discrete topology. 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 compact, even though its component spaces are.

### Convergence in the box topology

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 topology

The basis sets in the product topology have almost the same definition as the above, *except* with the qualification that *all but finitely many* *U _{i}* are equal to the component space

*X*. The product topology satisfies a very desirable property for maps

_{i}*f*:

_{i}*Y*→

*X*into the component spaces: the product map

_{i}*f*:

*Y*→

*X*defined by the component functions

*f*is continuous if and only if all the

_{i}*f*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.

_{i}## See also

## Notes

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

## References

- Steen, Lynn A. and Seebach, J. Arthur Jr.;
*Counterexamples in Topology*, Holt, Rinehart and Winston (1970). ISBN 0030794854. - Willard, Stephen (2004).
*General Topology*. Dover Publications. ISBN 0-486-43479-6.