# Star refinement

The general definition makes sense for arbitrary coverings and does not require a topology. Let ${\displaystyle X}$ be a set and let ${\displaystyle {\mathcal {U}}=(U_{i})_{i\in I}}$ be a covering of ${\displaystyle X}$, i.e., ${\displaystyle X=\bigcup _{i\in I}U_{i}}$. 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 ${\displaystyle U_{i}}$ that intersect ${\displaystyle S}$, i.e.:
${\displaystyle \mathrm {st} (S,{\mathcal {U}})=\bigcup {\big \{}U_{i}:i\in I,\ S\cap U_{i}\neq \emptyset {\big \}}.}$
The covering ${\displaystyle {\mathcal {U}}=(U_{i})_{i\in I}}$ of ${\displaystyle X}$ is said to be a refinement of a covering ${\displaystyle {\mathcal {V}}=(V_{j})_{j\in J}}$ of ${\displaystyle X}$ if every ${\displaystyle U_{i}}$ is contained in some ${\displaystyle V_{j}}$. 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 ${\displaystyle V_{j}}$. Finally, the covering ${\displaystyle {\mathcal {U}}}$ is said to be a star refinement of ${\displaystyle {\mathcal {V}}}$ if for every ${\displaystyle i\in I}$ the star ${\displaystyle \mathrm {st} (U_{i},{\mathcal {U}})}$ is contained in some ${\displaystyle V_{j}}$.