In this paper the following two propositions are proved: (a) If X-alpha, alpha is an element of A, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product square(alpha is an element of A) X-alpha can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If X-alpha, alpha is an element of A, is an infinite system of Brown Hausdorff topological spaces then the box product square X-alpha is an element of (A)alpha is also Brown Hausdorff, and hence, it is connected. A space is Brown if for every pair of its open nonempty subsets there exists a point common to their closures. There are many examples of countable Brown Hausdorff spaces in literature.