# 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 $\mathcal U$ if and only if:

$\forall V_\beta \in \mathcal V: \exists U_\alpha \in \mathcal U: V_\beta \subseteq U_\alpha$

That is, if and only if every element of $\mathcal V$ is the subset of some element of $\mathcal U$.

### Finer Cover

Let $\mathcal V$ be a refinement of $\mathcal U$.

Then $\mathcal V$ is finer than $\mathcal U$.

### Coarser Cover

Let $\mathcal V$ be a refinement of $\mathcal U$.

Then $\mathcal U$ 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.