Caret

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.:

$\mathrm{s}\mathrm{t}(S,\mathcal{𝒰})=\bigcup \{U_i:i\in I,S\cap U_i\ne \mathrm{\varnothing }\}.$

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

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 $\mathrm{s}\mathrm{t}(x,\mathcal{𝒰})$ 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 $\mathrm{s}\mathrm{t}(U_i,\mathcal{𝒰})$ 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.