Monotonically star compact spaces

A space X is monotonically star-compact   if one can assign to for each open cover UU a subspace s(U)⊆Xs(U)⊆X, called a kernel, such that s(U)s(U) is a compact subset of X  , and st(s(U),U)=Xst(s(U),U)=X, and if VV refines UU then s(U)⊆s(V)s(U)⊆s(V), where st(s(U),U)=⋃{U∈U:U∩s(U)≠∅}st(s(U),U)=⋃{U∈U:U∩s(U)≠∅}. In this paper, we investigate topological properties of monotonically star-compact spaces.