nLab box topology

Definition

Given a set $\{X_i\}_{i \in I}$ of topological spaces, then the box topology on the Cartesian product $\underset{i \in I}{\prod} X_i$ of the underlying sets of these spaces is the topology which is generated from the topological base whose elements are the Cartesian products

$$\underset{i \in I}{\prod} U_i \;\subset\; \underset{i\in I}{\prod} X_i$$

of open subsets $U_i \subset X_i$ of each of the factor spaces.

Properties

Relation to Tychonoff topology

If the index set $I$ is a finite set, then this box topology coincides with the Tychonoff topology on the Cartesian product. For general $I$ however the Tychnoff topology has as base open subsets only those products $\underset{i \in I}{\prod} U_i$ for which all but a finite number of factors are in fact the corresponding total space $X_i$.

Hence for non-finite index set $I$ the box topology is strictly finer that the Tychonoff topology. Beware that it is the Tychonoff toopology which yields the actual Cartesian product in the category Top of topological spaces. Accordingly, for non-finite $I$ the box topology fails to possess the properties that one expects from a categorical product.

For example, there may be a family of continuous functions, $f_i: X \to X_i$, for which the associated map by components, $X \to \underset{i\in I}{\prod} X_i$, is not continuous for the box topology.

References

• Wikipedia, Box topology