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 term has two similar but distinct usages. A related term sometimes used to differentiate the weaker of these two properties is the notion of a barycentric refinement. Star refinements are used in the definition of a fully normal space, and in one among several equivalent formulations of a uniform space.
Definitions 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,} that is, X = ⋃ U . {\textstyle X=\bigcup {\mathcal {U}}.} Given a subset S {\displaystyle S} of X , {\displaystyle X,} 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,} that is,
st ( S , U ) = ⋃ { U ∈ U : S ∩ U ≠ ∅ } . {\displaystyle \operatorname {st} (S,{\mathcal {U}})=\bigcup {\big \{}U\in {\mathcal {U}}:S\cap U\neq \varnothing {\big \}}.}
Given a point x ∈ X , {\displaystyle x\in X,} we write st ( x , U ) {\displaystyle \operatorname {st} (x,{\mathcal {U}})} instead of st ( { x } , U ) . {\displaystyle \operatorname {st} (\{x\},{\mathcal {U}}).}
A covering U {\displaystyle {\mathcal {U}}} of X {\displaystyle X} is 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 following are two special kinds of refinement. The covering U {\displaystyle {\mathcal {U}}} is called a barycentric refinement of V {\displaystyle {\mathcal {V}}} if for every x ∈ X {\displaystyle x\in X} the star st ( x , U ) {\displaystyle \operatorname {st} (x,{\mathcal {U}})} is contained in some V ∈ V . {\displaystyle V\in {\mathcal {V}}.} The covering U {\displaystyle {\mathcal {U}}} is called a star refinement of V {\displaystyle {\mathcal {V}}} if for every U ∈ U {\displaystyle U\in {\mathcal {U}}} the star st ( U , U ) {\displaystyle \operatorname {st} (U,{\mathcal {U}})} is contained in some V ∈ V . {\displaystyle V\in {\mathcal {V}}.}
A space X {\displaystyle X} is called fully normal if every open cover of X {\displaystyle X} has a barycentric open refinement.
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