Characterizations of ordinal spaces via continuous selections

We give several characterizations of ordinal spaces by means of the existence of a continuous selection which has the property that the value of every non-empty closed set is an isolated point of the closed set, and which has some additional properties. This is a generalization of result of Fujii and Nogura [Topology Appl. 91 (1999) 65–69], which characterizes compact ordinal spaces via continuous selections. Furthermore we give another simpler characterization of ordinal spaces via Fell-continuous selections.