Example 5

Let $(X_i)_{i\in I}$ and $(Y_j)_{j\in J}$ be families of B-spaces. Let X and Y be their sums. For each $(i, j)\in I\times J$, the mapping $(x, y)\mapsto$ $((i, x),(j, y))$ is a B-isomorphism of $X_i\times_BY_j$ onto the subspace $(\{i\} \times X_i)\times_B(\{j\} \times Y_j)$ of $X\times_BY$. Since the latter form a partition of $X\times_BY$ into open subsets, the mapping

defined by $h((i, j),(x, y)) = ((i, x),(j, y))$ is a B-isomorphism.