Here is an application of Theorem 46.11 to quotient maps. (Compare E-S57IX.)
Theorem. If \(p: A \to B\) is a quotient map and \(X\) is locally compact Hausdorff, then \(i_X \times p: X \times A \to X \times B\) is a quotient map.
Let \(Y\) be the quotient space induced by \(i_X \times p\); let \(q: X \times A \to Y\) be the quotient map. Show there is a bijective continuous map \(f: Y \to X \times B\) such that \(f \circ q = i_X \times p\).
Let \(g = f^{-1}\). Let \(G: B \to \mathcal{C}(X, Y)\) and \(Q: A \to \mathcal{C}(X, Y)\) be the maps induced by \(g\) and \(q\), respectively. Show that \(Q = G \circ p\).
Show that \(Q\) is continuous; conclude that \(G\) is continuous, so that \(g\) is continuous.
Solution
Let \(q:X\times A\to Y\) be the quotient induced by \(i_X\times p\).
The map \(i_X\times p\) is constant exactly on the fibers of \(q\), hence factors uniquely as \[ X\times A\xrightarrow qY\xrightarrow fX\times B. \] The induced \(f\) is bijective and continuous by the defining property of the quotient topology.
Let \(g=f^{-1}\). Give \(C(X,Y)\) the compact-open topology. Theorem 46.11 states that, for \(X\) locally compact Hausdorff and any space \(Z\), a map \(F:X\times Z\to Y\) is continuous if and only if the induced map \(\hat F:Z\to C(X,Y)\), \(\hat F(z)(x)=F(x,z)\), is continuous. The map \(g\) induces \[ G:B\to C(X,Y),\qquad G(b)(x)=g(x,b), \] and \(q\) induces \[ Q:A\to C(X,Y),\qquad Q(a)(x)=q(x,a). \] Since \(f q=i_X\times p\), equivalently \(q=g(i_X\times p)\), we have \[ Q(a)(x)=g(x,p(a))=G(p(a))(x), \] so \(Q=G\circ p\).
The map \(q:X\times A\to Y\) is continuous. Since \(X\) is locally compact Hausdorff, Theorem 46.11 says its adjoint \(Q:A\to C(X,Y)\) is continuous. Because \(p\) is quotient and \(Q=G\circ p\), the quotient criterion implies \(G\) is continuous. Applying Theorem 46.11 again, the adjoint \(g:X\times B\to Y\) is continuous. Hence the continuous bijection \(f\) has continuous inverse, so \(f\) is a homeomorphism. Therefore \(i_X\times p\) is the composite of the quotient map \(q\) with a homeomorphism, and is quotient.