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Let $\{ (X_\alpha,\T_\alpha) \}_{\alpha\in A}$ be a family of topological spaces. Let $Y$ denote the generalized Cartesian product of the sets $X_\alpha$ , that is$$ Y = \prod_{\alpha\in A} X_\alpha.$$ Let $\B$ denote the set of all products of open sets of the corresponding spaces, that is$$ \B = \left\{ \prod_{\alpha\in A} U_\alpha \,\Biggm|\, U_\alpha\in\T_\alpha \text{ for all } \alpha\in A \right\}.$$
Now we can construct the box product $(Y,\S)$ , where $\S$ , referred to as the box topology, is the topology generated by the base $\B$ .
When $A$ is a finite set, the box topology coincides with the product topology.
As an example, the box product of two topological spaces $(X_0,\T_0)$ and $(X_1,\T_1)$ is $(X_0\times X_1,\S)$ , where the box topology $\S$ (which is the same as the product topology) consists of all sets of the form $\bigcup_{i\in I}(U_i\times V_i)$ , where $I$ is some index set and for each $i\in I$ we have $U_i\in\T_0$ and $V_i\in\T_1$ .
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