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. A related concept is the notion of barycentric refinement. 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.

Definitions

The general definition makes sense for arbitrary coverings and does not require a topology. Let X be a set and let \mathcal U be a covering of X, that is, Given a subset S of X, the star of S with respect to \mathcal U is the union of all the sets that intersect S, that is, Given a point x \in X, we write instead of A covering \mathcal U of X is a refinement of a covering \mathcal V of X if every is contained in some The following are two special kinds of refinement. The covering \mathcal U is called a barycentric refinement of \mathcal V if for every x \in X the star is contained in some The covering \mathcal U is called a star refinement of \mathcal V if for every the star is contained in some

Properties and Examples

Every star refinement of a cover is a barycentric refinement of that cover. The converse is not true, but a barycentric refinement of a barycentric refinement is a star refinement. Given a metric space X, let be the collection of all open balls of a fixed radius \epsilon>0. The collection is a barycentric refinement of \mathcal V, and the collection is a star refinement of \mathcal V.