Definition:Precise Refinement of Cover

Definition
Let:


 * $\mathcal S = \left\{ {S_\gamma: \gamma \in \Gamma}\right\}$

be a cover of a set $X$.

Also let:


 * $\mathcal T = \left\{ {T_\gamma: \gamma \in \Gamma}\right\}$

be a cover of $X$.

Then $\mathcal T$ is a precise refinement of $\mathcal S$ :


 * $\forall \gamma \in \Gamma: T_\gamma \subseteq S_\gamma$