> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]], then $X$ is **paracompact** if every [[Open Cover|open cover]] has a locally finite refinement. > [!theorem] > > Let $(X, \topo)$ be a paracompact topological space, and $\seqi{U}$ be a locally finite open cover. Let $K \subset X$ be a compact set, then $K \cap U_i$ for finitely many $i \in I$. > > *Proof*. For each $x \in K$, there exists $V_x \in \cn^o(x)$ such that $V_x \cap U_i$ for finitely many $i \in I$. Since $\bracs{V_x}_{x \in K}$ forms an open cover of $K$, it has a finite subcover $\seqf{V_i}$. Hence $K \cap U_i$ for finitely many $i \in I$.