Definition:Refinement of Cover

Definition
Let $S$ be a set.

Let $\UU = \set {U_\alpha}$ and $\VV = \set {V_\beta}$ be covers of $S$.

Then $\VV$ is a refinement of $\UU$ :
 * $\forall V_\beta \in \VV: \exists U_\alpha \in \UU: V_\beta \subseteq U_\alpha$

That is, every element of $\VV$ is the subset of some element of $\UU$.

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

Also see

 * Definition:Subcover