Definition 3

The topological space $\prod_BX_i$ is called the fibred product of the family $(X_i)_{i\in I}$ over B. The B-space obtained by endowing $\prod_BX_i$ with the mapping $p$ is called the product B-space of the family $(X_i)_{i\in I}$.

Let $j\in I$. The mapping $(b, x)\mapsto$ pr$_j(x)$ of $\prod_BX_i$ into $X_j$ is called the projection of index $j$ of the fibred product and is again denoted by pr$_j$. It is continuous. It is a B-morphism, for one has $p=p_j\circ$ pr$_j$.

Let Y be a B-space and $q$ its projection. For each $i\in$ I, let $u_i: Y\rightarrow X_i$ be a B-morphism. There exists a unique B-morphism $v: Y\rightarrow$ $\prod_BX_i$ such that pr$_i\circ v=u_i$ for each $i\in I$ (universal property of the product B-space): it is the mapping of Y into $\prod_BX_i$, defined by $y\mapsto (q(y),(u_i(y))_{i\in I})$, which is sometimes denoted by $(u_i)_{i\in I}$.

Let $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ be families of B-spaces and, for each $i\in I$, let $f_i: X_i\rightarrow Y_i$ be a B-morphism. The mapping $(b,(x_i)_{i\in I})\mapsto$ $(b,(f_i(x_i))_{i\in I})$ is a B-morphism of $\prod_BX_i$ into $\prod_BY_i$ which is denoted by $\prod_Bf_i$ and is called the extension of the family $(f_i)_{i\in I}$ to the fibred products.