I know that the complement of the Cantor set can be written as the countable union of disjoint intervals i.e. $[0,1]\backslash K=\bigcup_{n\in\mathbb{N}}I_n$. However I was told that if $x\not\in \bigcup_{n\in\mathbb{N}}\overline{I_n}$, then there exists a $\delta >0$ such that if $I_n\cap(x-\delta, x +\delta)\neq \varnothing$ then $\overline{I_n}\subseteq(x-\delta,x+\delta)$. I'm not sure why this is true, and am starting to doubt whether it is in fact true.
Any clarification would be greatly appreciated!