Example 2

Let $(X, p)$ be a B-space and let $(A_k)_{k\in K}$ be a family of subspaces of B. Let A be the sum space of the family $(A_k)_{k\in K}$ and let Y be the sum space of the family $(\overset{-1}{p}(A_k))_{k\in K}$; let us denote by $i: A\rightarrow B$, $j: Y\rightarrow X$ and $p': Y\rightarrow A$ the mappings deduced from the canonical injections of $A_k$ into B, the canonical injections of $\overset{-1}{p}(A_k)$ into X, and the mappings $p_{A_k}:\overset{-1}{p}(A_k)\rightarrow A_k$, for $k\in K$. The square

Y $^j$ X

A $^i$ B is cartesian; this follows from the example of I, p. 10 and from prop. 6.