Star refinement

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

\mathrm {st} (S,{\mathcal {U}})=\bigcup {\big \{}U_{i}:i\in I,\ S\cap U_{i}\neq \emptyset {\big \}}.

Given a point $x\in X$ , we write $\mathrm {st} (x,{\mathcal {U}})$ instead of $\mathrm {st} (\{x\},{\mathcal {U}})$ .

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