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 be a set and let \( {\mathcal {U}} \) be a covering of \( {\displaystyle X=\bigcup _{U\in {\mathcal {U}}}(U)} \) . Given a subset S of X then the star of S with respect to \( {\mathcal {U}} \) is the union of all the sets \( {\displaystyle U\in {\mathcal {U}}} \) that intersect S, i.e.:
\( {\displaystyle \mathrm {st} (S,{\mathcal {U}})=\bigcup _{O\in {\big \{}U\in {\mathcal {U}}:U\cap S\neq \{\}{\big \}}}(O)}. \)
Given a point \( x\in X \), some authors write " \( {\displaystyle \mathrm {st} (x,{\mathcal {U}})} \)" instead of " \( {\displaystyle \mathrm {st} (\{x\},{\mathcal {U}})} \) ", although the former is an abuse of notation.
Note that \( {\displaystyle \mathrm {st} (S,{\mathcal {U}})\neq \bigcup _{O\in {\mathcal {U}}}(O\cap S)}. \)
The covering \( {\mathcal {U}} \) of } X is said to be a refinement of a covering \( {\mathcal {V}} \) of X iff \( {\displaystyle \forall U\in {\mathcal {U}},\,\exists V\in {\mathcal {V}}:U\subseteq V} \). The covering \( {\mathcal {U}} \) is said to be a barycentric refinement of \( {\mathcal {V}} \) iff \( {\displaystyle \forall x\in X,\,\exists V\in {\mathcal {V}}:\mathrm {st} (\{x\},{\mathcal {U}})\subseteq V} \) . Finally, the covering \( {\mathcal {U}} \) is said to be a star refinement of \( {\mathcal {V}} \) iff \( {\displaystyle \forall U\in {\mathcal {U}},\,\exists V\in {\mathcal {V}}:\mathrm {st} (U,{\mathcal {U}})\subseteq V}. \)
Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness.
References
J. Dugundji, Topology, Allyn and Bacon Inc., 1966.
Lynn Arthur Steen and J. Arthur Seebach, Jr.; 1970; Counterexamples in Topology; 2nd (1995) Dover edition ISBN 0-486-68735-X; page 165.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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