Prove Intersection of Two compact sets is compact using open cover?
Solution 1:
Let $A$ and $B$ be compact. Let $U=\{U_\alpha\}$ be an open covering of $A\cap B$. Since we are working in $\mathbb R$, we know that $A$ and $B$ are both closed. Thus, $A\cap B$ is closed, so $U'=U\cup \{\mathbb R\setminus (A\cap B)\}$ is an open cover of $A\cup B$. Now can you use this to construct a finite subcovering of $A\cap B$?