Box product-closed property of topological spaces

Definition
Suppose $$\alpha$$ is a property of topological spaces, i.e., for any topological space $$X$$, $$X$$ either satisfies $$\alpha$$ or does not satisfy $$\alpha$$. Then, we say that $$\alpha$$ is a box product-closed property of topological spaces if for any (possibly finite, possibly infinite) collection of topological spaces $$X_i, i \in I$$, all of which satisfy $$\alpha$$, the product space $$\prod_{i \in I} X_i$$, endowed with the box topology, also satisfies $$\alpha$$.

Weaker metaproperties

 * Stronger than::Finite product-closed property of topological spaces

Other related metaproperties

 * Product-closed property of topological spaces

To understand the relation between these concepts:


 * Examples of properties of topological spaces that are both product-closed and box product-closed: T1 space, Hausdorff space
 * Examples of properties that are box product-closed but not product-closed: discrete space
 * Examples of properties that are product-closed but not box product-closed: compact space, connected space. Basically this list includes properties that are about smallness or intimacy of some sort. The box topology makes separation easier.