I missed geometry in school, I'm trying to fill in the gaps by reading Pollack's Differential Topology. Am I doing this right?
This is #16 in the first section. Show the diagonal $\Delta$ of $X\times X$ is diffeomorphic to $X$.
All sets are subsets of some Euclidean space. I define $\phi:X\to X\times X$ by $\phi(x)=(x,x)$. Clearly $\phi$ is a bijection of $X$ and $\Delta$. If $x\in X$, let $U_x$ be an open nbhd of $x$ in the ambient space $\mathbb{R}^k$. Define $F:U_x\to\mathbb{R}^{2k}$ by $F(x)=(x,x)$. I suspect $F$ is smooth. Notice if $y\in U_x$, $$ \begin{align*} dF_y(z)&=\lim_{h\to 0}\frac{F(y+hz)-F(y)}{h}\\ &= \lim_{h\to 0}\frac{(y+hz,y+hz)-(y,y)}{h}\\ &=\lim_{h\to 0} (z,z)=F(z) \end{align*} $$ so $dF_y=F$. This implies $F$ is smooth. Since $F=\phi$ on $U_x\cap X$, $\phi$ is also smooth. Finally, $\phi^{-1}\colon\Delta\to X$ is just a projection map on the first coordinate, and I know projections are smooth. So $\phi$ is a diffeomorphism of $X$ and $\Delta$.
Is this sensible? Sorry, I don't have any place to get feedback. Thanks.