Definition:Precise Refinement of Cover

Definition
Let $\family{S_i}_{i \in I}$ be an indexed family of subsets of a set $S$ indexed by $I$.

Let:
 * $\SS = \set{S_i: i \in I}$

be a cover of $S$.

Also let $\family{T_i}_{i \in I}$ be an indexed family of subsets of $S$ indexed by $I$.

Let:
 * $\TT = \set {T_i: i \in I}$

be a cover of $S$.

Then $\TT$ is a precise refinement of $\SS$ :


 * $\forall i \in I: T_i \subseteq S_i$