Refinement of an open cover

A refinement of a cover $\mathcal U$ of a set $A\subseteq X$ is another cover $\mathcal V$ of $A$ such that for every $V\in\mathcal V$ there exists $U\in\mathcal U$ with

V \subseteq U.

If $\mathcal U$ and $\mathcal V$ are open covers, then $\mathcal V$ is called an open refinement of $\mathcal U$ .

Use with a basis

Refinements compare “how fine” two covers are and are especially useful when working with a basis of a topology, since open covers can often be refined by basic open sets.

Examples

In $\mathbb{R}$ , let $\mathcal U=\{(-1,1),(0,2)\}$ , which covers $[0,1]$ . Then $\mathcal V=\{(-1/4,3/4),(1/4,5/4)\}$ is a refinement of $\mathcal U$ .
If $\mathcal U$ is an open cover of $X$ and $\mathcal B$ is a basis, then taking all $B\in\mathcal B$ with $B\subseteq U$ for some $U\in\mathcal U$ gives an open refinement by basis elements.