$\sqcap_{\alpha}U_{\alpha}\cap\sqcap_{\alpha}A_{\alpha}=\sqcap_{\alpha}(U_{\alpha}\cap A_{\alpha})$ . On the left hand side we have a basis element for the subspace topology (where if $\prod_{\alpha}X_{\alpha}$ is in the product topology, for all but finitely many $\alpha$ , $U_{\alpha}=X_{\alpha}$ ), and on the right hand side we have a basis element for the topology of the product where each $A_{\alpha}$ is in its subspace topology (where, again, if the product is in the product topology, for all but finitely many $\alpha$ , we may assume $U_{\alpha}=X_{\alpha}$ so that $U_{\alpha}\cap A_{\alpha}=A_{\alpha}$ ). This shows that the bases for the two topologies (the subspace topology of the product and the topology of the product of subspace topologies) are the same, regardless of whether both products are given the box or product topology.