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 ${\displaystyle X}$ be a set and let ${\displaystyle {\mathcal {U}}}$ be a covering of ${\displaystyle X}$, i.e., ${\displaystyle X=\bigcup {\mathcal {U}}}$. Given a subset ${\displaystyle S}$ of ${\displaystyle X}$ then the star of ${\displaystyle S}$ with respect to ${\displaystyle {\mathcal {U}}}$ is the union of all the sets ${\displaystyle U\in {\mathcal {U}}}$ that intersect ${\displaystyle S}$, i.e.:

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

Given a point ${\displaystyle x\in X}$, we write ${\displaystyle \mathrm {st} (x,{\mathcal {U}})}$ instead of ${\displaystyle \mathrm {st} (\{x\},{\mathcal {U}})}$. Note that ${\displaystyle \mathrm {st} (S,{\mathcal {U}})\neq \bigcup _{O\in {\mathcal {U}}}(O\cap S)}$.

The covering ${\displaystyle {\mathcal {U}}}$ of ${\displaystyle X}$ is said to be a refinement of a covering ${\displaystyle {\mathcal {V}}}$ of ${\displaystyle X}$ if every ${\displaystyle U\in {\mathcal {U}}}$ is contained in some ${\displaystyle V\in {\mathcal {V}}}$. The covering ${\displaystyle {\mathcal {U}}}$ is said to be a barycentric refinement of ${\displaystyle {\mathcal {V}}}$ if for every ${\displaystyle x\in X}$ the star ${\displaystyle \mathrm {st} (x,{\mathcal {U}})}$ is contained in some ${\displaystyle V\in {\mathcal {V}}}$. Finally, the covering ${\displaystyle {\mathcal {U}}}$ is said to be a star refinement of ${\displaystyle {\mathcal {V}}}$ if for every ${\displaystyle U\in {\mathcal {U}}}$ the star ${\displaystyle \mathrm {st} (U,{\mathcal {U}})}$ is contained in some ${\displaystyle V\in {\mathcal {V}}}$.

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.