Example 3

Let X be a B-space and $p$ its projection. The fibred product $X\times_BX$ of X and X over B is called the fibred square of X. It is the subspace of $X\times X$ formed by the pairs $(x, x')$ such that $p(x) =p(x')$. It is endowed with the structure of a B-space whose projection is the mapping $(x, x')\mapsto$ $p(x)$. The diagonal $\Delta_X$ of $X\times X$ (E, II, p. 13) is contained in $X\times_BX$; it is also called the diagonal of $X\times_BX$. The mapping $x\mapsto (x, x)$ of X into $X\times_BX$ is a B-morphism, called the diagonal B-morphism and often denoted by $\delta_X$; it defines a B-isomorphism of X onto $\Delta_X$ (TG, I, p. 25, cor. 2).