Posets having continuous intervals (Q1434352)

The authors consider posets in which each order interval \([a,b]\) is a continuous poset or continuous domain. The main result: If \(X\) is a core compact space and \(L\) is a poset equiped with the Scott topology (assumed to satisfy a mild extra condition) for which each interval is a continuous sup-semilattice, then the function space of continuous locally bounded functions from \(X\) into \(L\) has intervals that are continuous sup-semilattices. The authors give an example of a distributive lattice such that its lattice of Scott-open sets is not continuous.