Star refinement

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In mathematics, in the study in topology of 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 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 U_i.

For each x we can take the union of the sets in its star, say V_x. The condition on an open cover

W_j

for j in J to be a star refinement of the U_i is then

1. each W_j is contained in some U_i (refinement condition), and
2. each set taken V_x with respect to the W_j is contained in a U_i (star condition).