How to show that the diagonal of $X\times X$ is diffeomorphic to $X$? 
The diagonal $Q$ in $X\times X$ is the set of points of the form $(x,x)$. Show that $Q$ is diffeomorphic to $X$, so $Q$ is a manifold if $X$ is.

Can anyone please help me to solve this question I have been trying to solve it since two days but I could not.
Thanks.
 A: Since the term diffeomorphism applies by definition only to smooth manifods, the statement - as stated - makes little sense. Instead one can show the following
Proposition 1. Let $X$ be a topological space and $Q\subseteq X\times X$ the diagonal (endowed with the subspace topology of the product topology). Then the canonical embedding $i\colon x\mapsto (x,x)$ is a homeomorphism $X\to Q$.
Proof.
The map $i$ is the unique continuous map $X\to X\times X$ that the universal property of the direct product guarantees for the identity $X\to X$ on both factors. On the other hand, the restriction to $Q$  of the canonical projection of $X\times X\to X$ (for either factor) is obviously inverse to $i$. $_\square$
In other words, $Q$ has exactly the same topological properties as $X$ has, especially
Corollary. $Q$ is a manifold if and only $X$ is. $_\square$
Proposition 2. Assume $X$ is a smooth manifold. Then the homeomorphism $i$ from proposition 1 is a smooth embedding (and hence $i$ is a diffeomorphism $X\to Q$).
Proof.
Let $n=\dim X$, so $\dim(X\times X)=2n$.
It suffices to show that for each $(x,x)\in Q$ there exists a chart $\phi\colon U\to \mathbb R^{2n}$ in the smooth atlas of $X\times X$ (as obtained for the product space from the given smooth atlas of $X$) such that  $(x,x)\in U\subseteq X\times X$ and $\phi(U\cap Q)$ is the intersection of an $n$-dimensional plane with $\phi(U)$.
To do so, pick a chart $\psi\colon V\to \mathbb R^n$ from the smooth atlas of $X$ for which $x\in V$. Then $U:=V\times V$ is open in $X\times X$ and by the definition of smooth structure on $X\times X$, the map $\phi:=\psi\times \psi\colon U\to \mathbb R^{2n}=\mathbb R^n\times \mathbb R^n$, $(\xi,\eta)\mapsto (\psi(\xi),\psi(\eta))$ belongs to the smooth atlas of $X\times X$. As $\phi(U\cap Q)$ is the diagonal plane $\mathbb R^n\times \mathbb R^n$, the claim follows. $_\square$
