# 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 , we write instead of .

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.

## 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 ISBN 0-486-68735-X; page 165.