\begin{definition}[Definition:Precise Refinement of Cover]
Let:
:$\SS = \set {S_\gamma: \gamma \in \Gamma}$
be a cover of a set $X$.
{{explain|What is $\Gamma$? Is this an indexing set? If so, this needs to be noted.}}
Also let:
:$\TT = \set {T_\gamma: \gamma \in \Gamma}$
be a cover of $X$.
Then $\TT$ is a '''precise refinement''' of $\SS$ {{iff}}:
:$\forall \gamma \in \Gamma: T_\gamma \subseteq S_\gamma$
Category:Definitions/Covers
\end{definition}