\begin{definition}[Definition:Open Refinement]
Let $T = \struct {X, \tau}$ be a topological space.
Let $\UU$ and $\VV$ be covers of $X$.
Then $\VV$ is an '''open refinement''' of $\UU$ {{iff}}:
:$(1): \quad \forall V \in \VV: \exists U \in \UU: V \subseteq U$
:$(2): \quad \VV \subseteq \tau$
That is:
:$(1): \quad \VV$ is a refinement of $\UU$
:$(2): \quad$ All elements of $\VV$ are open in $T$
Category:Definitions/Covers
\end{definition}