Countability and star covering properties

Whenever P is a topological property, we say that a topological space is star P if whenever U is an open cover of X, there is a subspace A ⊆ X with property P such that X = St ( A , U ) . We study the relationships of star P properties for P ∈ { Lindelöf , σ -compact, countable } with other Lindelöf type properties.