Definition of product topology/box topology in terms of the Kuratowski closure operator?

Question

In Munkres, the definition of box topology is given in terms of open sets. I am wondering if there is a definition of the box topology/product topology in terms of the Kuratowski closure operator? i.e. how do we define the kuratowski closure operator of $A \times B$ given those given on $A$ and $B$?

Answer 1

$$\mathrm{Cl}_{\mathcal{T}_{\square}}(A\times B)=\mathrm{Cl}_{\mathcal{T}_\times}(A\times B)=\mathrm{Cl}_{\mathcal{T}_0}(A)\times\mathrm{Cl}_{\mathcal{T}_1}(B)$$ where $\mathcal{T}_{\square}$ is the box topology and $\mathcal{T}_\times$ is the product topology on $X\times Y$ and $\mathcal{T}_0$ is the topology on $X\supseteq A$ and $\mathcal{T}_1$ is the topology on $Y\supseteq B$.