Proposition 10

Let X be a topological space and let R be an equivalence relation in X. Let Y be a locally compact topological space. Let S be the equivalence relation in $X\times Y$ which is the product of the equivalence relation R in X and of the equality relation in Y. The canonical bijection $(X\times Y)/S\rightarrow (X/R)\times Y$ is a homeomorphism.

Recall that if U and V are topological spaces, $\mathscr{C}_c(U; V)$ denotes the set of continuous mappings of U into V, endowed with the topology of compact convergence (TG, X, p. 26, def. 1).

Let $p: X\rightarrow X/R$ and $q: X\times Y\rightarrow (X\times Y)/S$ be the canonical surjections. Let $g: (X\times Y)/S\rightarrow (X/R)\times Y$ denote the canonical bijection. It is continuous; let $h$ denote its inverse and let us prove that it is continuous.

The mapping $i: X\rightarrow \mathscr{C}_c(Y; X\times Y)$ such that, for every $x\in$ X, $i(x)$ is the mapping defined by $y\mapsto (x, y)$, is continuous (TG, X, p. 28, th. 3). The mapping $\widetilde{q}:\mathscr{C}_c(Y; X\times Y)\rightarrow \mathscr{C}_c(Y; (X\times Y)/S)$ which associates with a continuous mapping $\varphi$ the mapping $q\circ \varphi$, is continuous (TG, X, p. 29, prop. 9). Consequently, the mapping $\widetilde{q}\circ i: X\rightarrow \mathscr{C}_c(Y; (X\times Y)/S)$ is continuous. It is compatible with the equivalence relation R; the unique mapping $j: X/R\rightarrow \mathscr{C}_c(Y; (X\times Y)/S)$ such that $j\circ p=\widetilde{q}\circ i$ is therefore continuous. We have $h(\xi , y) =j(\xi )(y)$ for every pair $(\xi , y)\in (X/R)\times Y$. Since Y is locally compact, it then follows from TG, X, p. 28, th. 3, that the mapping $h$ is continuous.

The conclusion of Proposition 10 is no longer necessarily true if Y is not locally compact (TG, I, p. 96, exerc. 6).