Definition:Star Refinement

Definition
Let $S$ be a set.

Let $\CC$ be a cover for $S$.

Let $\VV$ be a cover for $S$ such that:
 * $ \forall x \in S: \exists U \in \CC: x^* \subseteq U$

where $x^*$ is the star of $x$ with respect to $\VV$.

That is:
 * $\displaystyle x^* := \bigcup \set {V \in \VV: x \in V}$

the union of all sets in $\VV$ which contain $x$.

Then $\VV$ is a star refinement of $\CC$.