let $X$ be a $n$-manifold. let $A=\{(x,y,z) \, |\,x=y\}$. I want to see if $A$ is a submanifold of $X^3$.
Consider the map $\Delta\times 1:X\times X\rightarrow X \times X\times X;\, (x,y)\mapsto (x,x,y)$.
If $U_x$ and $U_y$ are neighborhoods of $x$ and $y$ in $X$ then $\Delta\times 1 (U_x\times U_y)=\Delta(U_x)\times U_y$ is a neighborhood of $\Delta(x,y)=(x,x,y)$ in $X^3$, and we can see that $\Delta(U_x)\times U_y$ is in $A$. So this is a neighborhood of $(x,x,y)$ in $A$.
Now $U_x\cong \mathbb R^n$ and $\Delta(U_x)\cong U_x\cong \mathbb R^n$ and so $A$ is an $2n$-manifold.
Is my argument correct?and does it matter that the diagonal $\Delta(U_x)$ is a closed set, I mean don't we always want a neighborhood to be open?