Box Topological Products of Topological Spaces

# Box Topological Products of Topological Spaces

Recall that if $\{ X_i \}_{i \in I}$ is an arbitrary collection of topological spaces and $\displaystyle{\prod_{i \in I} X_i}$ is the Cartesian product of these spaces then we can define the product topology on $\displaystyle{\prod_{i \in I} X_i}$ to be the topology $\tau$ induced by the collection of projection maps $\displaystyle{\prod_{i \in I} \to X_i}$. We then said that the resulting topological space is a topological product.

Given an arbitrary collection of topological spaces $\{ X_i \}_{i \in I}$ there is another topology we can put on the product $\displaystyle{\prod_{i \in I} X_i}$ known as the box topology which we define below.

 Definition: Let $\{ X_i \}_{i \in I}$ be an arbitrary collection of topological spaces and let $\displaystyle{\prod_{i \in I} X_i}$ be the Cartesian product. Then the Box Topology on $\displaystyle{\prod_{i \in I} X_i}$ is the topology $\tau$ with a basis $\mathcal B = \left \{ \prod_{i \in I} U_i : U_i \subseteq X_i \: \mathrm{is \: open \: for \: all \:} i \in I \right \}$. The space $\displaystyle{\prod_{i \in I} X_i}$ with the box topology is called a Box Topological Product or simply Topological Product if the context of the topology is unambiguous.

Sometimes we write $\displaystyle{\prod_{i \in I}^{\mathrm{BOX}} X_i}$ to denote the space $\displaystyle{\prod_{i \in I} X_i}$ is accompanied with the box product topology.

In other words, open sets in a box topological product $\displaystyle{\prod_{i \in I} X_i}$ are sets $U = \prod_{i \in I} U_i$ where each set $U_i$ in the Cartesian product is an open set in the corresponding topological space $X_i$.

It is important to note that if $\{ X_i \}_{i \in I}$ is a finite collection of topological spaces then the product topology and box product topology on $\displaystyle{\prod_{i \in I} X_i}$ produce the same topological space.