> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]] and $A \subseteq X$. A collection of [[Open Set|open sets]] in the [[Relative Topology|induced topology]] of $A$, $\Sigma \subseteq \topo$ form an **open cover** of $A$ if $A \subset \bigcup_{U \in \Sigma}U$. > [!definition] > > Let $\Sigma$ be an open cover of $X$. $\Sigma$ is **locally finite** if for each $x \in X$, there exists a [[Neighbourhood|neighbourhood]] $U \in \cn^o(x)$ such that $U$ only intersects finitely many sets in $\Sigma$. > [!definition] > > Let $\mathcal{U}, \mathcal{V}$ be open covers of $X$. $\mathcal{V}$ is a **refinement** of $\mathcal{U}$ if for every $V \in \mathcal V$ there exists $U \in \mathcal U$ such that $V \subset U$.