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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 $X$ be a set and let $𝒰 = (U_{i})_{i \in I}$ be a covering of $X$ , i.e., $X = ⋃_{i \in I} U_{i}$ . Given a subset $S$ of $X$ then the star of $S$ with respect to $𝒰$ is the union of all the sets $U_{i}$ that intersect $S$ , i.e.:

s t (S, 𝒰) = ⋃ {U_{i} : i \in I, S \cap U_{i} \neq \emptyset} .

Given a point $x \in X$ , we write $s t (x, 𝒰)$ instead of $s t ({x}, 𝒰)$ .

The covering $𝒰 = (U_{i})_{i \in I}$ of $X$ is said to be a refinement of a covering $𝒱 = (V_{j})_{j \in J}$ of $X$ if every $U_{i}$ is contained in some $V_{j}$ . The covering $𝒰$ is said to be a barycentric refinement of $𝒱$ if for every $x \in X$ the star $s t (x, 𝒰)$ is contained in some $V_{j}$ . Finally, the covering $𝒰$ is said to be a star refinement of $𝒱$ if for every $i \in I$ the star $s t (U_{i}, 𝒰)$ is contained in some $V_{j}$ .

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.

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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>
+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]]

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