From the KKM theorem for the “closed” and “open” valued cases, we deduce a generalization of the Alexandroff–Pasynkoff theorem, existence theorems for almost fixed points of lower semicontinuous multimaps, and a partial solution of the Ben-El-Mechaiekh conjecture.

Any strict partial order R on a nonempty set X defines a function θR which associates to each strict partial order S ⊆ R on X the strict partial order θR(S) = R ◦ S on X. Owing to the strong relationships between Alexandroff TD derivative operators and strict partial orders, this paper firstly calls forth the links between the CantorBendixson ranks of Alexandroff TD topological spaces and the g...