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