====== 2020-02-10, Monday ======
$$\gdef\wt\widetilde \gdef\RM\backslash$$

===== Whitney Approximation Theorem. =====
**Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F:  M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$. 

Sketch of the proof: We do it following steps
  - By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$
  - For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/2, \z{ and } \delta(x)/2 < \delta(y)  \}$. 
  - The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover $\{U_{x_i}\}_{i=1}^\infty$, and set $U_i = U_{x_i}$. 
  - Do a partition of unity $\{\varphi_0, \varphi_i\}$ subject to $\{U_i\}_{i=0}$. Then define the function as $$ \wt F(x) = F_0(x) \varphi_0(x) + \sum_{i=1}^\infty F(x_i) \varphi_i(x) $$ Claim, this function works. 

===== Tubular Neighborhood Theorem =====
The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small. In order to do this, we first embed $M$ to $\R^m$ for some big $m$,
$$ \iota: M \into \R^m $$
 then we smooth the composition $\iota \circ F: N \to M$, to get $\wt G: N \to \R^m$, $C^0$-close to the original image of $\iota(M)$. Finally, we project $\wt G(N)$ back onto $M$. This smoothing-then-project-back operation gives a smooth map from $N$ to $M$.