In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. The general definition makes sense for arbitrary coverings and does not require a topology. Let be a set and let be a covering of that is, Given a subset of then the star of with respect to is the union of all the sets that intersect that is, Given a point we write instead of Note that The covering of is said to be a refinement of a covering of if every is contained in some The covering is said to be a barycentric refinement of if for every the star is contained in some Finally, the covering is said to be a star refinement of if for every the star is contained in some Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness. See also References J. Dugundji, Topology, Allyn and Bacon Inc., 1966. Lynn Arthur Steen and J. Arthur Seebach, Jr.; 1970; Counterexamples in Topology; 2nd (1995) Dover edition ; page 165. Topology Families of sets