Lemma 1

*Let X be a locally compact space, R an open equivalence relation in X, such that the quotient space X/R is paracompact; let π be the canonical mapping of X onto X/R. There exists a continuous real-valued function F ≥ 0 on X such that:

a) F is not identically zero on any equivalence class with respect to R;

b) for every compact subset K of X/R, the intersection of $\overline{\pi}(K)$ with Supp F is compact.

To each point $u \in X/R$ let us associate a function $f_u \in \mathscr{K}_+(X)$ such that $f_u$ is not identically zero on $\overline{\pi}(u)$; let $\Omega_u$ be the open set of points where $f_u > 0$; thus $u \in \pi(\Omega_u)$. Since $\pi$ is an open mapping, the $\pi(\Omega_u)$ form an open covering of X/R. There exists a locally finite open covering $(U_\iota)_{\iota \in I}$, finer than the covering by the $\pi(\Omega_u)$, then (GT, IX, §4, No. 3, Prop. 3) a partition of unity $(g_\iota)_{\iota \in I}$ on X/R subordinate to the covering $(U_\iota)$. For every $\iota \in I$, choose a $u_\iota$ such that $U_\iota \subset \pi(\Omega_{u_\iota})$. The function $F_\iota = (g_\iota \circ \pi) \cdot f_{u_\iota}$ belongs to $\mathscr{K}(X)$ and has support contained in $\overline{\pi}(U_\iota)$. The supports of the $F_\iota$ therefore form a locally finite family, so that one can define a continuous function $F \geq 0$ on X by setting $F = \sum_{\iota \in I} F_\iota$.

For every $u \in X/R$, there exists an $\iota$ such that $g_\iota(u) > 0$ and therefore $u \in U_\iota$; next, there exists an $x \in \Omega_{u_\iota}$ such that $\pi(x) = u$; then $f_{u_\iota}(x) > 0$ and $g_\iota(\pi(x)) > 0$, therefore $F_\iota(x) > 0$ and a fortiori $F(x) > 0$; this proves that F has the property a). Finally, let K be a compact subset of X/R. There exists a finite subset J of I such that, for $\iota \in I - J$, one

Same as TG, IX, §4, n° 4, prop. 5 (absent from GT).

has $U_\ell \cap K = \varnothing$, therefore $\overline{\pi}^{-1}(K) \cap \mathrm{Supp}\ F_\ell = \varnothing$. Then the set

is compact.