Let $X$ be a topological space and $S_3$ the symmetric group acting on $X^3$ by permuting coordinates.
Let $\pi:X^3\rightarrow X^3/S_3$. Denote $[x,y,z]=\pi(x,y,z)$. Let $U_x$ be the neighborhood of $x$ in $X$. Then $(U_x\times U_y\times U_z)/S_3$ is a neighborhood of $[x,y,z]$.
Now consider the point $[x,x,z]\in X^3/S_3$ i.e., $(x=y)$, why a neighborhood of $[x,x,z]$ is $(U_x\times U_x)/S_2 \times U_z$ instead of $(U_x\times U_x\times U_z)/S_3$?