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In [[mathematics]], specifically in the study of [[topology]] and [[open cover]]s 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 <math>X</math> be a set and let <math>\mathcal U=(U_i)_{i\in I}</math> be a covering of <math>X</math>, i.e., <math>X=\bigcup_{i\in I}U_i</math>. Given a subset <math>S</math> of <math>X</math> then the ''star'' of <math>S</math> with respect to <math>\mathcal U</math> is the union of all the sets <math>U_i</math> that intersect <math>S</math>, i.e.:


: <math>\mathrm{st}(S,\mathcal U)=\bigcup\big\{U_i:i\in I,\ S\cap U_i\ne\emptyset\big\}. </math>
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Given a point <math>x\in X</math>, we write <math>\mathrm{st}(x,\mathcal U)</math> instead of <math>\mathrm{st}(\{x\},\mathcal U)</math>.
 
The covering <math>\mathcal U=(U_i)_{i\in I}</math> of <math>X</math> is said to be a ''refinement'' of a covering <math>\mathcal V=(V_j)_{j\in J}</math> of <math>X</math> if every <math>U_i</math> is contained in some <math>V_j</math>. The covering <math>\mathcal U</math> is said to be a ''barycentric refinement'' of <math>\mathcal V</math> if for every <math>x\in X</math> the star <math>\mathrm{st}(x,\mathcal U)</math> is contained in some <math>V_j</math>. Finally, the covering <math>\mathcal U</math> is said to be a ''star refinement'' of <math>\mathcal V</math> if for every <math>i\in I</math> the star <math>\mathrm{st}(U_i,\mathcal U)</math> is contained in some <math>V_j</math>.
 
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 ==
* [[James Dugundji|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.
 
[[Category:Topology]]

Latest revision as of 09:59, 3 December 2014


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