Exercise 2, § 9

Let X be a separated manifold of class $C^r(1\leq r\leq \infty )$, countable at infinity, and pure of dimension $n$.

a) Assume that there exists an embedding $\varphi$ of X into a finite dimensional vector space V. Show that there exists an embedding of X in $\mathbf{R}^{2n+1}$ (if dim $V>2n+ 1$, prove that there exists a point $p$ of V such that, for all $x\in X$, the straight line joining $p$ to $\varphi (x)$ meets $\varphi (V)$ only in the point $\varphi (x)$, and meets it transversally at this point; deduce that there is an embedding of X in a space of dimension equal to dim $V-1$).

b) Show that there exists an embedding of X in $\mathbf{R}^{2n+1}$. (Let $\mathscr{O}$ be the set of open subsets of $X,\mathscr{U}$ the subset of $\mathscr{O}$ consisting of the open sets U such that there exists a morphism $\varphi : X\rightarrow \mathbf{R}^{2n+1}$ whose restriction to U is an embedding; by using $a$), show that $\mathscr{U}$ is a quasi-full subset (General Topology, Chap. IX, §4, Exerc. 27) of $\mathscr{O}$, and hence is equal to $\mathscr{O}.)$

c) Show that there exists a proper embedding of X in $\mathbf{R}^{2n+1}$. (Construct a proper embedding of X in $\mathbf{R}^{2n+2}$ by means of a proper function on X.)