Understanding a theorem of Whitney While watching a  video (min 34)on differentiable manifolds the lecturer mentioned a theorem of Whitney which sounded striking to me. The theorem is as follows 
Any $C^k{\geq1}$ atlas of a topological manifold contains a $C^\infty$ atlas.
What exactly the theorem says? Can one elaborate a little bit on this? 
 A: The theorem says that if we are given an $n$-dimensional topological manifold $M$, and if $X$ is a $C^k$ atlas for $M$ with $k \ge 1$, then there exists a subset $Y \subset X$ such that $Y$ is a $C^\infty$ atlas. 
Just to peel this apart a little bit, each element of $X$ is a coordinate chart $U \xrightarrow{f_U} \mathbb{R}^n$, where $U \subset M$ is open and $f_U$ is a homeomorphism onto an open subset of $\mathbb{R}^n$ (if you like to sprinkle a little formality on your notation, you may write this element of the atlas $X$ as an ordered pair $(U,f_U) \in X$). To say that $X$ is a $C^k$ atlas for $M$ means that if $U \xrightarrow{f_U} \mathbb{R}^n$ and $V \xrightarrow{f_V} \mathbb{R}^n$ are two elements of $X$ then the overlap function $f_U(U \cap V) \xrightarrow{f_U^{-1}} U \cap V \xrightarrow{f_V} f_V(U \cap V)$ is a $C^k$ diffeomorphism between the two open subsets $f_U(U \cap V)$ and $f_V(U \cap V)$ of $\mathbb{R}^n$.
So the meaning of the theorem is that if $X$ is a $C^k$ atlas, then by carefully picking and choosing amongst the coordinate charts that are the elements of the atlas $X$ and throwing away all of the rest, the charts that are left will form a smaller atlas $Y$ having the stronger property that the overlap maps between any two of them are $C^\infty$.
