Lemma 1

Let T and $T'$ be two topological spaces, A and $A'$ compact subsets of T and $T'$, respectively, W a neighbourhood of $A\times A'$ in $T\times T'$. There exists an open neighbourhood U of A in T and an open neighbourhood $U'$ of $A'$ in $T'$ such that $U\times U'\subset W$.

Let $x\in A$; there exist open subsets $U_x$ of T and $U'_x$ of $T'$ such that $\{x\} \times A'\subset U_x\times U'_x\subset W:$ indeed, the compact subset $\{x\} \times A'$ of $T\times T'$ can be covered by a finite number of open sets contained in W, of the form $U_i\times U'_i$, with $x\in U_i$; it suffices to take $U_x=\bigcap_iU_i$ and $U'_x=\bigcup_iU'_i$.

Since A is compact, there exist points $x_1, . . . , x_m$ of A such that $A\subset \bigcup_iU_{x_i}$; put $U =\bigcup_iU_{x_i}$ and $U'=\bigcap_iU'_{x_i}$. Then $A\times A'\subset U\times U'\subset W$, hence the lemma.

In the remainder of this number, Y denotes a separated manifold of class $C^r$.