Example 3

When the set I is a set with one element $\alpha$ (resp. with two elements $\alpha$ and $\beta$; resp. with three elements $\alpha ,\beta ,\gamma$ ), the mapping pr$_{\alpha}$ (resp. (pr$_{\alpha}$, pr$_{\beta}$) ; resp. (pr$_{\alpha}$, pr$_{\beta}$, pr$_{\gamma}$)) of $\prod_BX_i$ into $X_{\alpha}$ (resp. into $X_{\alpha}\times_BX_{\beta}$; resp. into $X_{\alpha}\times_BX_{\beta}\times_BX_{\gamma}$) is a B-isomorphism. This will allow us to deduce the properties of the fibred product of two or three B-spaces from those of the fibred product of families of B-spaces.

Let $(X_i)_{i\in I}$ be a family of B-spaces and let J be a subset of I. From the mapping Id$_B\times$ pr$_J$ of $B\times \prod_{i\in I}X_i$ into $B\times \prod_{i\in J}X_i$, by passing to subsets, one deduces a B-morphism $\prod_{i\in IB}X_i\rightarrow \prod_{i\in JB}X_i$. One

again denotes it by pr$_J$ and calls it the projection of index J of the fibre product.

Let $(X_i)_{i\in I}$ be a family of B-spaces. Let $(J_{\lambda})_{\lambda\in L}$ be a partition

of I. The mapping (pr$_{J_{\lambda}}$)$_{\lambda\in L}$ of $\prod_{i\in IB}X_i\rightarrow \prod_{\lambda\in LB}(\prod_{i\in J_{\lambda}B}X_i)$ is a

B-isomorphism ("associativity" of fibre products of B-spaces).