A note on isolated points of the compact superkernel of compact spaces

In this paper, we study some properties of the set of isolated points of the compact superkernel of compact spaces. For a T_1-space, we prove that I(\lambda X)\subset {\lambda _c}X; i.e., each isolated point in \lambda X must be a compact maximal linked system. Moreover, for an infinite compact space X with I(X)=\emptyset, we prove that I(\lambda X) =\emptyset; i.e., if X is a pointless space, then its superextension \lambda X is also pointless.