Proposition 4

Let A be a compact subset of X. There exist a finite dimensional vector space E and a morphism $\varphi \in \mathscr{C}^r(X;E) (r\not=\omega )$ that is an embedding in the neighbourhood of A.

Let $(U_i, \varphi_i,E_i)_{i\in I}$ be a finite family of charts of X whose domains cover A. We extend $\varphi_i$ to a map from X to $E_i$ (also denoted by $\varphi_i$) by putting $\varphi_i(x) = 0$ for $x \notin U_i$. Let $(V_i)_{i\in I}$ be a covering of A by open subsets of X such that $\overline{V}_i\subset U_i$ for all $i\in I$ (the existence of such a covering follows from General Topology, Chap. IX, §4, no. 3, Cor. 1 of Th. 3, applied to the compact space $X'$ obtained by adjoining to X a point at infinity and the covering of $X'$ consisting of the open sets $U_i(i\in I)$ and $X'$ -- A). For all $i\in I$, let $\alpha_i$ be a numerical function of class $C^r$ on X, equal to 1 at every point of $V_i$, and with support contained in $U_i($Differentiable and Analytic Manifolds, Results, 5.3.6).

Consider the map $\varphi : X\rightarrow \bigoplus_{i\in I}(E_i\oplus \mathbf{R})$ defined by

For all $i\in I$, the map $\alpha_i\varphi_i$ is of class $C^r$ (since its restrictions to $U_i$ and to the complement of the support of $\alpha_i$ are), and its restriction to $V_i$ is an embedding; it follows that $\varphi$ is a morphism of class $C^r$ and is an immersion at every point of A. We show that the restriction of $\varphi$ to A is injective. Let $x, y$ be two points of A such that $\varphi (x) =\varphi (y)$, and let $i\in I$ be such that $x\in V_i$. Then $\alpha_i(x) = 1$, so $\alpha_i(y) = 1$, which implies that $y\in U_i$; but we also have $\varphi_i(x) =\varphi_i(y)$, so $x=y$ since $\varphi_i$ induces an embedding of $U_i$ into $E_i$.

It can be shown$^9$ that every separated manifold, countable at infinity and of

pure dimension $n$, embeds in $\mathbf{R}^{2n}$; for a weaker result, cf. Exercise 2.

$^9$ See H. WHITNEY, The self-intersections of a smooth $n$-manifold in $2n$-space,

Ann. of Math., 45 (1944), pp. 220-246.