Proposition 2

Assume that the manifold Y is paracompact; let A be a subset of X, and let $\varphi :X\rightarrow Y$ be a morphism of class $C^r$ that induces a homeomorphism from A to $\varphi (A)$, and that is étale at every point of A. Then there exists an open neighbourhood U of A such that $\varphi$ induces an isomorphism from U to an open submanifold of Y.

Restricting X and Y if necessary, we can assume that $\varphi$ is étale and surjective. Denote by $\sigma :\varphi (A)\rightarrow A$ the inverse homeomorphism of $\varphi |A$. Since Y is metrizable (Differentiable and Analytic Manifolds, Results, 5.1.6), $\varphi (A)$ admits a fundamental system of paracompact neighbourhoods; hence, by General Topology, Chap. XI, there exist an open neighbourhood V of $\varphi (A)$ in Y and a continuous map $s: V\rightarrow X$, that coincides with $\sigma$ on $\varphi (A)$ and is such that $\varphi (s(y)) =y$ for all $y\in V$. Moreover, $s$ is topologically étale, so $s(V)$ is an open set U containing A. Then $\varphi$ induces a homeomorphism $\varphi '$ from U to V; by Differentiable and Analytic Manifolds, Results$, 5.7.8,\varphi '$ is an isomorphism.

In the remainder of this number, we assume that $r\not=\omega$.