The product of any basis elements is open in the box topology, and the product of finitely many basis elements and all other spaces is open in the product topology. Hence, the topologies defined in the theorem are coarser than the box and product topologies, respectively. Further, $\prod_{\alpha}U_{\alpha}=\cup_{B_{\alpha}\subset U_{\alpha}}\prod_{\alpha}B_{\alpha}$ where each $B_{\alpha}$ is a basis element if $U_{\alpha}$ is a proper subset of $X_{\alpha}$ or $X_{\alpha}$ if $U_{\alpha}=X_{\alpha}$ . Hence, every open basis element for either topology can be represented as a union of some basis elements as defined in the theorem, and the topologies defined in the theorem are finer than the box and product topologies, respectively.