High Quality Content by WIKIPEDIA articles! 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 1. each Ui is contained in some Vj (the refinement condition), and 2. each star U*(x) is contained in a Vj (star condition). That is, 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 .,

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