Supremum of an intersection of sets
We do not always have equality. For example let: \begin{eqnarray*}X_1&=&[0,1] \sqcup [3,4],\\X_2&=&[0,2].\end{eqnarray*} so sup$X_1=4$, sup$X_2=2$, but sup$X_1\cap X_2=1$.
If the $X_i$ are all connected, and the intersection of them is non-empty, then we have equality, simply because the intersection is then an interval with infimum the supremum of the infima of the $X_i$, and supremum the infimum of the suprema of the $X_i$.