Definition:Precise Refinement of Cover

Definition

Let:

$\SS = \set {S_\gamma: \gamma \in \Gamma}$

be a cover of a set $X$.

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Also let:

$\TT = \set {T_\gamma: \gamma \in \Gamma}$

be a cover of $X$.

Then $\TT$ is a precise refinement of $\SS$ if and only if:

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