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