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., ${\displaystyle X=\bigcup _{U\in {\mathcal {U}}}(U)}$. 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.:

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

Given a point ${\displaystyle x\in X}$, some authors write "${\displaystyle \mathrm {st} (x,{\mathcal {U}})}$" instead of "${\displaystyle \mathrm {st} (\{x\},{\mathcal {U}})}$", although the former is an abuse of notation.

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 ${\displaystyle X}$ iff ${\displaystyle \forall U\in {\mathcal {U}},\,\exists V\in {\mathcal {V}}:U\subseteq V}$. The covering ${\displaystyle {\mathcal {U}}}$ is said to be a barycentric refinement of ${\displaystyle {\mathcal {V}}}$ iff ${\displaystyle \forall x\in X,\,\exists V\in {\mathcal {V}}:\mathrm {st} (\{x\},{\mathcal {U}})\subseteq V}$. Finally, the covering ${\displaystyle {\mathcal {U}}}$ is said to be a star refinement of ${\displaystyle {\mathcal {V}}}$ iff ${\displaystyle \forall U\in {\mathcal {U}},\,\exists V\in {\mathcal {V}}:\mathrm {st} (U,{\mathcal {U}})\subseteq V}$.

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.