Confusion: Cannot understand the statement that $(0,1)^2$ cannot be represeted as union of countable open balls.

Question

In sharp contrast to the claim in Union of a countable collection of open balls, we have the following assertion in Christopher Heil's book Introduction to Real Analysis

Is this an error on part of the author? Please help I am confused both of these statements (including the one in the link on Math.SE) do not seem to be true at the same time

Answer 1

If a subset $U\subseteq \mathbb R^n$ is the union of (no matter how many, but at least two) disjoint open balls, then $U$ is necessarily disconnected.

If $n=1$, then every bounded connected open subset of $\mathbb R^n$ is already an open ball but the same is not true if $n\geq2$.