Update

Signed-off-by: Marcello Seri <[email protected]>

6-differentiaforms.tex

@@ -916,7 +916,7 @@ \section{De Rham cohomology and Poincar\'e lemma}
 In fact this is the case, thanks to the following theorem\footnote{This is a deep result related to the Whitney Embedding Theorem from Remark~\ref{rmk:WhitneyET} and is out of the scope of our course, for more details refer to~\cite[Chapter 6 and Theorems 6.26 and 9.27]{book:lee}.}.
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 \begin{theorem}[Whitney Approximation Theorem for continuous maps]\label{thm:WhitneyApproxCont}
-Given any continuous mapping $G \in C^0(M,N)$, there exists $F \in C^\infty(M,N)$ which is homotopic to $G$. Moreover, if $G$ is smooth\footnote{Note that a function $f : M \to N$ is defined to be smooth on a subset $A \subset M$ if there is some smooth function $g: U \to N$, defined on an open $U\supset A$ such that $g = f$ on $A$.} on a closed subset $U\subset M$, then one can choose $F$ so that $F=G$ on $U$.
+Given any continuous mapping $G \in C^0(M,N)$, there exists $F \in C^\infty(M,N)$ which is homotopic to $G$. Moreover, if $G$ is smooth\footnote{Note that a function $f : M \to N$ is defined to be smooth on a subset $A \subset M$ if there is some smooth function $g: U \to N$, defined on an open $U\supset A$ such that $g = f$ on $A$.} on a closed subset $A\subset M$, then one can choose $F$ so that $F=G$ on $A$.
 \end{theorem}
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 In particular, if two smooth maps are homotopic then they are also smoothly homotopic: we can assume the map $K$ to be smooth.