Star Refinement
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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_{...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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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