Definition:Star Refinement

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Let $S$ be a set.

Let $\mathcal C$ be a cover for $S$.

Let $\mathcal V$ be a cover for $S$ such that:

$ \forall x \in S: \exists U \in \mathcal C: x^* \subseteq U$

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

That is:

$\displaystyle x^* := \bigcup \left\{{V \in \mathcal V: x \in V}\right\}$

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

Then $\mathcal V$ is a star refinement of $\mathcal U$.