Proving a Compact Hausdorff manifold can be embedded in $\mathbb{R}^q$

Question

I have been reading Hirsch's Book Differential Topology and here he claims that

Let $M$ be a compact Hausdorff manifold of class $C^r$, $1\leq r\leq \infty$. Then there exists a $C^r$ embedding of $M$ into $\mathbb{R}^q$ for some $q$.

Now I only have one doubt in this proof that is in the beggining he claims that if we have charts $\{(\phi_i,U_i)\}_{i=1}^m$ that cover $M$, by composing with a diffeomorphism if necessary we can assume that $D(2)\subset \phi_i(U_i)$ and that $M=\cup_{i=1}^m int(\phi_i^{-1}(D(1)))$. Now I get each of the conditions independently but I can't seem to make both work at the same time, especially in the case that we have a chart $U_j$ that is disjoint from every other chart. Any help clarifying this is appreciated. Thanks in advance.

Answer 1

For this kind of argument, it's useful to start with altogether too many charts and avoid invoking compactness until the end.

• Choose a charts around each point $\{(\varphi_p,U_p):p\in M\}$, with $p\in U_p$, translated and scaled so that $\varphi_p(p)=0$ and $D(2)\subset\varphi_p(U_p)$.
• Let $V_p=\varphi_p^{-1}(D(1))$. Note that the set $\{V_p:p\in M\}$ is an open cover of $M$
• Choose a finite subcover $V_{p_1},\dots,V_{p_m}$ of this cover.

The charts $\{(\varphi_{p_i},U_{p_m}):i\in [1,m]\}$ have the desired properties.