Prove that there are two compact subsets $K_1$ and $K_2$ of $X$ such that $K = K_1 \cup K_2, K_1 \subset U_1, K_2 \subset U_2$.

Question

Let $X$ be a Hausdorff space, let $K$ be a compact subset of $X$, and let $U_1,U_2$ be an open sets of $X$ such that $K\subset U_1\cup U_2$. Prove that there are two compact subsets $K_1$ and $K_2$ of $X$ such that $K = K_1 \cup K_2, K_1 \subset U_1, K_2 \subset U_2$.

Hello, could someone guide me to solve this problem, please? Best regards.

Answer 1

$K\setminus U_1$ and $K\setminus U_2$ are disjoint compact sets. By a standard theorem we can find disjoint open sets $V_1$ and $V_2$ such that $K\setminus U_1 \subset V_1$ and $K\setminus U_2 \subset V_2$. You can take $K_1=K\setminus V_1$ and $K_2=K\setminus V_2$. Let me know if you need more details or reference for the theorem I have used.