showing compactness of the intersection of a compact set and disjoint open sets. Here's the problem I'm trying to solve.
"If $A\subseteq B_1\cup B_2$ where $B_1, B_2$ are disjoint open sets and $A$ is compact,     show that $A\cap B_1$ is compact. Is the same true if $B_1$ and $B_2$ are not disjoint?"
Hope you can help, I can't seem to wrap my head around this one. Thanks!

Let $\{V_{\alpha}\}_{\alpha\in J}$ be a family of open sets such that:
$$A\cap B_1\subseteq\cup_{\alpha\in J}V_{\alpha}$$Since $A\subseteq B_1\cup B_2$, it follows that $A\subseteq\ B_2\cup(\cup_{\alpha\in J}V_{\alpha})$. Since $A$ is compact , therefore there is a finite subset of $\{{V_{\alpha}}|\alpha\in J\}\cup\{B_2\}$ that covers $A$. Now try to use this subset (along with the fact that $B_1\cap B_2=\emptyset$) to make a finite subset of $\{{V_{\alpha}}|\alpha\in J\}$ that covers $A\cap B_1$.
