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.:

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

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.