In this paper we consider posets in which each order interval [a,b] is a continuous poset or continuous domain. After developing some basic theory for such posets, we derive our major result: if X is a core compact space and L is a poset equipped 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. This substantially generalizes known results for continuous domains. © 2004 Elsevier B.V. All rights reserved.
Lawson, J. D., & Xu, L. (2004). Posets having continuous intervals. In Theoretical Computer Science (Vol. 316, pp. 89–103). https://doi.org/10.1016/j.tcs.2004.01.025