Example 4

Let $(B_i)_{i\in I}$ be a family of topological spaces and let, for each $i\in I$, $(X_i, p_i)$ and $(Y_i, q_i)$ be $B_i$-spaces. Put $B =\prod_{i\in I}B_i$, $X =\prod_{i\in I}X_i$ and $Y =\prod_{i\in I}Y_i$. Endowed with the continuous mapping $p=\prod_ip_i$ (resp. $q=\prod_iq_i$), the topological space X (resp. Y) is a B-space. By the associativity isomorphism of the topological products of $\prod_i(X_i\times Y_i)$ onto $(\prod_iX_i)\times (\prod_iY_i) = X\times Y$ (TG, I, p. 25, prop. 2), the subspace $\prod_i(X_i\times_{B_i}Y_i)$ of $\prod_i(X_i\times Y_i)$ is identified with $X\times_BY$.