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 .
- J. Dugundji, Topology, Allyn and Bacon Inc., 1966.