Definition 2

The topological space $X\times_BX'$ is called the fibred product of X and $X'$ over B. The B-space obtained by endowing $X\times_BX'$ with the mapping $q$ is called the product B-space of X and $X'$.

The restrictions to $X\times_BX'$ of the projections of $X\times X'$ into X and into $X'$ are again denoted by pr$_1$ and pr$_2$ and called the first and second projections of the fibred product. They are continuous and are B-morphisms, for one has $q=p\circ$ pr$_1=p'\circ$ pr$_2$.

It should be noted that $X\times_BX'$ may be empty even if X and $X'$ are non-empty: in fact, the relation $X\times_BX'=\emptyset$ is equivalent to saying that $p(X)$ and $p'(X')$ are disjoint.

Let Y be a B-space and $u: Y\rightarrow X,u': Y\rightarrow X'$ B-morphisms. There exists a unique B-morphism $v: Y\rightarrow X\times_BX'$ such that pr$_1\circ v=u$ and pr$_2\circ v$ = $u'$ (universal property of the product B-space of two B-spaces): this is the mapping $y\mapsto (u(y), u'(y))$ of Y into $X\times_BX'$, which is sometimes denoted by $(u, u')$.

Let $X,X',Y,Y'$ be B-spaces, let $f: X\rightarrow Y,f': X'\rightarrow Y'$ be B-morphisms. The mapping $(x, x')\mapsto (f(x), f'(x'))$ is a B-morphism of $X\times_BX'$ into $Y\times_BY'$, which is denoted by $f\times_Bf'$ and is called the extension of $f$ and $f'$ to fibred products.