Monotone properties defined from stars of open coverings

We define some monotone properties using stars of coverings. This relates to work of J. van Mill, V. Tkachuk, R. Wilson, O. Alas, L. Junqueira, M. Matveev and others who generalized the D-space property of E. van Douwen and E. Michael. Given a property P , we call a topological space X monotonically star- P if one can assign to each open cover U a subspace s ( U ) ⊆ X with property P in such a way that st ( s ( U ) , U ) = ⋃ { U ∈ U : U ∩ s ( U ) ≠ ∅ } = X and if V refines U then s ( U ) ⊆ s ( V ) . We study monotonically star- P spaces for various compactness-like properties P .