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.

Given x in X and an open cover {Ui} of X with index set I, the star of x with respect to the cover is the set of i in I such that x is in Ui. That is,
 * $$ x^* = \{i \in I \;|\; x \in U_i\} .\,$$

For each x we can take the union of the sets in its star; this is also called the star of x. That is,
 * $$ U^*(x) = \bigcup_{i \in x^*} U_i .\,$$

Then this open cover is a star refinement of some other open cover {Vj}, with index set J, if That is, Actually, the star condition alone is enough; the refinement condition follows (except in a degenerate case when X is the empty set).
 * 1) each Ui is contained in some Vj (the refinement condition), and
 * 2) each star U*(x) is contained in a Vj (star condition).
 * 1) $$ \forall i \in I,\; \exists j \in J,\; U_i \subseteq V_j ,\,$$
 * 2) $$ \forall x \in X,\; \exists j \in J,\; U^*(x) \subseteq V_j .\,$$

Star refinements are used in the definition of fully normal space and in one definition of uniform space.