Definition:Refinement of Cover

Definition
Let $S$ be a set.

Let $\mathcal U = \left\{{U_\alpha}\right\}$ and $\mathcal V = \left\{{V_\beta}\right\}$ be covers of $S$.

Then $\mathcal V$ is a refinement of $\mathcal U$ iff:
 * $\forall V_\beta \in \mathcal V: \exists U_\alpha \in \mathcal U: V_\beta \subseteq U_\alpha$

That is, iff every element of $\mathcal V$ is the subset of some element of $\mathcal U$.

Finer Cover
If $\mathcal V$ is a refinement of $\mathcal U$, then:


 * $\mathcal V$ is finer than $\mathcal U$, and


 * $\mathcal U$ is coarser than $\mathcal V$.

Note
Although specified for the cover of a set, a refinement is usually used in the context of a topological space.