Let $L$ be a Dedekind-complete Banach lattice. Let $\mathcal{B}$ be the family of nonempty norm-compact subsets of $L$ that are bounded from below. Endow $\mathcal{B}$ with the topology induced by the Hausdorff metric. 


Consider the function $\psi\colon \mathcal{B}\to L$ defined by $\psi(A)= \inf{(A)}$ for all $A\in \mathcal{B}$.


I have a number of questions on this, the most important of which is the following:


*Q1. What conditions on $L$ guarantee that $\psi$ is a continuous function?*


The other questions depend on the answer to Q1, in obvious ways.


*Q2. What is an example of a Dedekind-complete Banach lattice for which $\psi$ is not continuous?*


*Q3 Suppose that $\psi$ is continuous and $E$ is a Banach lattice. Will the same hold for the the space of regular operators $\mathcal{L}^r(E,L)$ (the linear span of positive operators)? How about when $L$ has a strong order dual?* 


Now suppose that $X\subseteq L$ is a non-empty norm-compact sub-lattice of $L$. Take the restriction of $\psi$ to the space $\mathcal{B}(X)$ of nonempty compact subsets of $X$. My final question is the following:


*Q4 What conditions on $X$ guarantee that $\psi$ restricted to $\mathcal{B}(X)$ is continuous?*


This latter question Q4 is the same as asking, What conditions guarantee that $X$ as a topological lattice has a neighborhood base of $inf$-sub-semilattices?



Continuity of lattice operations in Banach lattices
I guess finite dimensionality is needed for Q2. For example, in $\ell\_2$ consider the sets $X\_n = \{n^{-1/2} e\_j : 1 \le j \le n \}$. Then in the Hausdorff metric $X\_n \to \{0\}$ but the infimum of each $X\_n$ has norm one. 


It looks like you need to restrict attention to compact sets that are in a common order interval.


