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 , i.e., . Given a subset of then the star of with respect to is the union of all the sets that intersect , i.e.: . Given a point , some authors write "" instead of "", although the former is an abuse of notation. Note that . The covering of is said to be a refinement of a covering of iff . The covering is said to be a barycentric refinement of iff . Finally, the covering is said to be a star refinement of iff . 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.