\begin{definition}[Definition:Refinement of Cover]
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 $\UU$ {{iff}}:
:$\forall V_\beta \in \VV: \exists U_\alpha \in \UU: V_\beta \subseteq U_\alpha$
That is, {{iff}} every element of $\VV$ is the subset of some element of $\UU$.
\end{definition}