A closed set $E$ without holes ( holes means bounded component of its complement) is said to be Arakelian if for every disc $D$ ( or compact sets in general) the union of all holes of $E \cup D $ are bounded.
A closed set $E$ without holes is Arakelian iff $\mathbb{C}\cup \{\infty\}\backslash E$ is locally connected at $\infty$ .
How we can prove that these two definitions are equivalent.