Supremum infinum notation

Question

I came across the following question: Let $\Omega$ denote the set of closed subsets of $[0,1]$. Show that $\rho : \Omega \times \Omega \to \mathbb{R}$ defined by $$\rho(A,B) := \max\left\{\sup_{x\in A}\inf_{y \in B}\vert x - y \vert, \sup_{y \in B}\inf_{x \in A}\vert x - y \vert\right\}.$$ I just want to know what this is supposed to mean. How can you take the infinum of $\vert x - y \vert$ over $y \in B$ before choosing an $x \in A$? And it makes no sense to take the supremum and infinum at the same time.

Answer 1

Essentially you just read it from left to right as $$ \sup_{x\in A}\left(\inf_{y\in B}|x-y|\right). $$ In other words, to compute $\sup_{x\in A}\inf_{y\in B}|x-y|$, we first find $$ \alpha_x:=\inf_{y\in B}|x-y| $$ for each $x\in A$ and then take $\sup_{x\in A}\alpha_x$.