Let K be a non-empty compact subset of $\mathbb R$ . Prove that $K$ is of the form $[a,b]$ or of the form $$[a,b] - \bigcup\limits_{n} I_n $$ where $I_n $ is a countable disjoint family of open intervals with end points in $K$.
Will this statement be true?
My Try : I think this statement is true. As $K$ will be contained in any closed interval $[a,b]$ where $a , b \in K$. and $K$ will be a closed So $\mathbb R - K$ is the union of countable union of disjoint open intervals . $\mathbb R - K = \bigcup\limits_{n} I_n$.$\;$ So $K = [a,b] - \bigcup\limits_{n} I_n $
Am I correct? Can anyone please mention where did I wrong?