One of problems of the book "Differentiable Manifolds" by Brickell & Clark is as follows (page 38):
Show that a discrete topological space cannot be given a $C^{\infty}$ structure.
Here it is what i did:
Suppose that a set $M$ is a discrete topological space and that we then give it a $C^{\infty}$ structure. We know that this atlas induces a topological structure. Let $x:U\subset M \to V\subset\mathbb{R}^n$ be any chart of the given atlas. $x$ is a homeomorphism relative to the discrete topology. Because for any $a\in V$, the set $\{a\}$ is open in the discrete topology. Since $x$ is injective, $x^{−1}(a)$ is a single element of $U$ and $\{x^{−1}(a)\}$ is open in the discrete topology on $M$. Hence $\{x\}$ is continuous. Similarly, it can been shown that $x$ is an open map. Hence it is a homeomorphism.Therefore induced topology on $M$ coincides with the discrete topology. This is where i have stopped !
Thanks in advance for any clarification.