Definition 4

One says that square (1) is a Cartesian square of topological spaces (or, simply, that it is Cartesian) if it is commutative and if, for every topological space Y and every pair of continuous mappings $u: Y\rightarrow B',v: Y\rightarrow X$ such that $f\circ u=p\circ v$, there exists a unique continuous mapping $w: Y\rightarrow X'$ such that $p'\circ w=u$ and $f'\circ w=v$.

For square (1) to be Cartesian, it is necessary and sufficient that the square

X $^p$ B

be Cartesian.