Proposition 3

Let A be a compact subset of X. The set $\mathscr{P}$ of morphisms $\varphi \in \mathscr{C}^r(X;Y)$ that are embeddings in the neighbourhood of A is open in $\mathscr{C}^r(X;Y)$ in the topology of compact $C^r$-convergence (§ 6, no. 4).

Clearly, it suffices to prove the proposition for $r= 1$.

a) We show first that the subset J of $\mathscr{C}^1(X;Y)$ consisting of the morphisms that are immersions at every point of A is open. Consider the map $j_A:\mathscr{C}^1(X;Y)\times A\rightarrow J^1(X;Y)$ such that $j_A(\varphi , x) =j_x^1(\varphi ) ($Differentiable and Analytic Manifolds, Results, 12.1).

By definition of the topology on $\mathscr{C}^1$(X;Y), the map $\widetilde{j}_A:\varphi \rightarrow j_A(\varphi , .)$ from $\mathscr{C}^1(X;Y)$ to $\mathscr{C}(A;J^1(X;Y))$ is continuous; it now follows from General Topology, Chap. X, §3, no. 4, Th. 3, that $j_A$ is continuous.

On the other hand, let M be the set of jets $j$ in $J^1(X;Y)$ whose tangent map $T(j) : T_{\mathbf{s}(j)}(X)\rightarrow T_{\mathbf{b}(j)}(Y) ($Differentiable and Analytic Manifolds, Results, 12.3.4) is injective. The set M is open in $J^1(X;Y)$; indeed, it suffices to verify this assertion when X is an open subset of a finite dimensional vector space E, and Y is an open subset of a Banach space F; we are then reduced (Differentiable and Analytic Manifolds, Results, 12.3.1) to proving that the set of injective continuous linear maps is open in $\mathscr{L}$(E;F), which follows from Spectral Theory, Chap. III, §2, no. 7, Prop. 16.

We conclude from the preceding that the set $j_A^{-1}(M)$ is open in $\mathscr{C}^1(X;Y)\times A$, hence that its complement $\mathscr{F}$ is closed. Since A is compact, the projection pr$_1:\mathscr{C}^1(X;Y)\times A\rightarrow \mathscr{C}^1(X;Y)$ is a proper morphism, hence closed; consequently, the set J, which is equal to $\mathscr{C}^1(X;Y)$ -- pr$_1(\mathscr{F})$, is open in $\mathscr{C}^1(X;Y)$.

b) Let H be the subset of $J\times A\times A$ consisting of the elements $(f, x, y)$ such that $f(x) =f(y)$. It is clear that H contains $J\times \Delta$, where $\Delta$ denotes the diagonal in the product $A\times A$; we show that $H'= H$ -- $(J\times \Delta )$ is closed in $J\times A\times A$. Since $\mathscr{P}$ is the complement in J of the image of $H'$ under the proper projection pr$_1: J\times A\times A\rightarrow J$, this will imply the proposition.

The topology of $\mathscr{C}^1(X;Y)$ being finer than the topology of compact convergence, the map $(\varphi , x) \rightarrow \varphi (x)$ from $\mathscr{C}^1(X;Y)\times A$ to Y is continuous (General Topology, Chap. X, §3, no. 4, Cor. 1 of Th. 3); it follows that H is closed in $J\times A\times A$. Hence, it suffices to show that $J\times \Delta$ is open in H, in other words, that for all $\varphi \in J$ and all $x\in A$ there exists a neighbourhood $\Omega$ of $\varphi$ in J and a neighbourhood B of $x$ in X such that, for any morphism $\psi$ in $\Omega$, the restriction of $\psi$ to $A\cap B$ is injective.

Thus, the proposition follows from the following lemma: