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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.
The general definition makes sense for arbitrary coverings and does not require a topology. Let X {\displaystyle X} be a set and let U {\displaystyle {\mathcal {U}}} be a covering of X {\displaystyle X} , i.e., X = ⋃ U {\textstyle X=\bigcup {\mathcal {U}}}. Given a subset S {\displaystyle S} of X {\displaystyle X} then the star of S {\displaystyle S} with respect to U {\displaystyle {\mathcal {U}}} is the union of all the sets U ∈ U {\displaystyle U\in {\mathcal {U}}} that intersect S {\displaystyle S} , i.e.:
Given a point x ∈ X {\displaystyle x\in X} , we write st {\displaystyle \operatorname {st} } instead of st {\displaystyle \operatorname {st} }. Note that st ≠ ⋃ O ∈ U {\textstyle \operatorname {st} \neq \bigcup _{O\in {\mathcal {U}}}}.
The covering U {\displaystyle {\mathcal {U}}} of X {\displaystyle X} is said to be a refinement of a covering V {\displaystyle {\mathcal {V}}} of X {\displaystyle X} if every U ∈ U {\displaystyle U\in {\mathcal {U}}} is contained in some V ∈ V {\displaystyle V\in {\mathcal {V}}}. The covering U {\displaystyle {\mathcal {U}}} is said to be a barycentric refinement of V {\displaystyle {\mathcal {V}}} if for every x ∈ X {\displaystyle x\in X} the star st {\displaystyle \operatorname {st} } is contained in some V ∈ V {\displaystyle V\in {\mathcal {V}}}. Finally, the covering U {\displaystyle {\mathcal {U}}} is said to be a star refinement of V {\displaystyle {\mathcal {V}}} if for every U ∈ U {\displaystyle U\in {\mathcal {U}}} the star st {\displaystyle \operatorname {st} } is contained in some V ∈ V {\displaystyle V\in {\mathcal {V}}}.