Let \(f\colon X\to Z\) and \(g\colon Y\to Z\) be continuous maps of topological spaces. The pullback of \(f\) and \(g\) is the subspace \[ X \cross_Z Y \coloneqq \ts{ (x,y) \in X\cross Y \st f(x) = g(y) } \] of the product space \(X\cross Y\), together with the restrictions \(p_X\colon X\cross_Z Y\to X\) and \(p_Y\colon X\cross_Z Y\to Y\) of the two projections.
Let \(f\colon X\to Z\) and \(g\colon Y\to Z\) be continuous maps with pullback \((X\cross_Z Y, p_X, p_Y)\). Then \(f\circ p_X = g\circ p_Y\), and for every topological space \(W\) and continuous maps \(a\colon W\to X\) and \(b\colon W\to Y\) with \(f\circ a = g\circ b\) there is a unique continuous map \(u\colon W\to X\cross_Z Y\) with \(p_X\circ u = a\) and \(p_Y\circ u = b\), namely \(u(w) = (a(w), b(w))\). Thus \(X\cross_Z Y\) is the limit of the diagram \(X\xrightarrow{f} Z \xleftarrow{g} Y\) in \(\Top\). If \(Z\) is a point, then \(X\cross_Z Y = X\cross Y\).
[@Hat02].