Definition 3.12.4 (Refinement).label Let $X$ be a topological space and $\cali{U},\cali{V}\suf\cali{P}(X)$ open covers then $\cali{V}$ is a refinement of $\cali{U}$ if for every $V\in\cali{V}$, there exists $U\in\cali{U}$ such that $V\suf U$.
Definition 3.12.4 (Refinement).label Let $X$ be a topological space and $\cali{U},\cali{V}\suf\cali{P}(X)$ open covers then $\cali{V}$ is a refinement of $\cali{U}$ if for every $V\in\cali{V}$, there exists $U\in\cali{U}$ such that $V\suf U$.
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