Definition 4.18.4 (Refinement). Let $X$ be a topological space and $\mathcal{U}, \mathcal{V}\subset 2^{X}$ be open covers, then $\mathcal{V}$ is a refinement of $\mathcal{U}$ if for every $V \in \mathcal{V}$, there exists $U \in \mathcal{U}$ such that $V \subset U$.