Let $X$ be a Hausdorff space, and $K\subset X$ compact, and $U_1,U_2\subset X$ open sets, and suppose that $K\subset U_1\cup U_2$, show that exist $K_1\subset U_1$ and $K_2\subset U_2$ compact in $X$ such that $K=K_1\cup K_2$ and $K_1\cap K_2=\emptyset$.
I was able to almost prove it, if $U_1\cap U_2=\emptyset$, then I can use that K is $T_4$, and use the propertie that $K\setminus U_1$ and $K\setminus U_2$ are disjoint compact sets because they are closed subsets of $K$, and then define $K\setminus U_2=K_1$, and $K\setminus U_1=K_2$.
Then for the case that $U_1$ and $U_2$ are not disjoint I've tried to follow a tip from the exercise, I've found open sets $V_1$ and $V_2$, that $K\setminus U_1\subset V_1$ and $K\setminus U_2\subset V_2$, and then set $K_1=K\setminus V_1$ and $K_2=K\setminus V_2$. But then things become a mess, and I could not prove the result from that. So I would be glad if anyone could help to finish this proof.
Thanks in Advance.