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 ${\displaystyle X,}$ that is, ${\textstyle X=\bigcup {\mathcal {U}}.}$ 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 ${\displaystyle S,}$ that is,

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

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

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 ${\displaystyle \operatorname {st} (x,{\mathcal {U}})}$ 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 ${\displaystyle \operatorname {st} (U,{\mathcal {U}})}$ 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.