Definition:Refinement of Cover/Finer Cover

Definition
Let $S$ be a set.

Let $\mathcal U = \left\{{U_\alpha}\right\}$ and $\mathcal V = \left\{{V_\beta}\right\}$ be covers of $S$. Let $\mathcal V$ be a refinement of $\mathcal U$.

Then $\mathcal V$ is finer than $\mathcal U$.