The property B(P,∝)-refinability is studied and is used to obtain new covering characterizations of paracompactness, collectionwise normality, subparacompactness, d-paracompactness, a-normality, mesocompactness, and related concepts. These new characterizations both generalize and unify many well-known results. The property B(P,∝)-refinability is strictly weaker than the property Θ-refinability. A B(P,∝)-refinement is a generalization of a σ-locally finite-closed refinement. Here ∝ is a fixed ordinal which dictates the number of "levels" in a given refinement, and P represents a property such as discreteness or local finiteness which each "level" must satisfy relative to a certain subspace. / Ph. D. / incomplete_metadata