Definition:Precise Refinement of Cover

Definition
Let:


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

be a cover of a set $X$.

Also let:


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

be a cover of $X$.

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


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