On monotonically star countably compact spaces

A space X is said to be monotonically star countably compact if one assign to for each open cover U a subspace s(U) subset of X, called a kernel, such that s(U) is a countably compact subset of X and St(s(U); U) = X, and if V refines U then s(U) subset of s(V), where St(s(U); U) = boolean OR{U is an element of U : U boolean AND s(U) not equal (empty set}. In this paper, we investigate the relationships between monotonically star countably compact spaces and other related spaces, and also study the topological properties of monotonically star countably compact spaces.