Supremum of an intersection of sets

Question

Let $(X_i)_{i\in I}$ be a collection of bounded subsets of $\mathbb{R}$.\ It's easy to show that $\bigcap_{i\in I} X_i$ is bounded (when it's non-empty) and that : \begin{equation} \sup\left( \bigcap_{i\in I}X_i\right) \leq \inf_{i\in I} \sup X_i \end{equation} My question is whether we have equality? And if not, in what conditions do we have it?

Answer 1

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 $X_i$.