Refinement

Definition
Let $$U_i, i \in I$$ be a cover of a topological space $$X$$, viz a collection of subsets of $$X$$ whose union is $$X$$. A refinement of $$U_i$$ is another cover $$V_j, j \in J$$ such that for any $$j$$, there exists an $$i$$ such that $$V_j \subseteq U_i$$.

An open refinement is a refinement where the new cover is an open cover.

Related notions

 * Shrinking
 * Expansion