Generalized Helly spaces, continuity of monotone functions, and metrizing maps
Given an ordered metric space (in particular, a Banach lattice) $E$, the generalized Helly space $H(E)$ is the set of all increasing functions from the interval $[0,1]$ to $E$ considered with the topology of pointwise convergence, and $E$ is said to have property $(\lambda)$ if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces $C(K,E)$, where $K$ is a compact space, that have property $(\lambda)$. In doing so, the guiding idea comes from the fact that there is a natural one-to-one correspondence between increasing functions $f:[0,1]\to C(K,E)$ (with countably many discontinuities) and continuous maps $F:K\to H(E)$ (with metrizable ranges). It leads to the investigation of general continuous metrizing maps (those with metrizable ranges), and especially of the so called separately metrizing maps, and the results obtained are then used to derive some permanence properties of the class of spaces $C(K,E)$ with property $(\lambda)$. For instance, it is shown that if $K$ is the product of compact spaces $K_j$ $(j\in J)$ such that each of the spaces $C(K_j,E)$ has property $(\lambda)$, so does $C(K,E)$; and, for any compact space $K$, if both $C(K)$ and a Banach lattice $E$ have property $(\lambda)$, so does $C(K,E)$.