Continuity of function spaces from pointwise directed families of characteristic functions. (Q890017)

The paper extends and refines some techniques introduced in the investigation of \textit{X. Xi} and \textit{J. Liang} [Topology Appl. 156, No. 3, 542-548 (2009; Zbl 1158.06004)]. The main result states that the function space \([X\to Q]\) is a continuous dcpo provided that \(X\) is locally compact and coherent and \(Q\) is a retract of a bifinite domain with \(\bot\). The crucial method used to establish continuity of the function space is to construct for a specific function \(f\) a family of characteristic functions with the property that for any \(x\in X\), the family of characteristic functions evaluated at \(x\) gives a directed subset of \(Q\) with supremum \(f(x)\). Such a family of functions need no longer be directed as a family of functions, but it has sufficient directed structure to ensure continuity.