Star refinement
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.
