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!
Best Answer
Hint: Let $\cal U$ be an open cover of $A\cap B_1$ then $\{U\cup B_2\mid U\in\cal U\}$ is an open cover of $A$.
For the second part note that $[0,1]\subseteq (-1,1)\cup(\frac12,\frac43)$.