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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

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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

Iconoclast
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    The key word being *disjoint*. – Lee Mosher Jun 26 '22 at 17:27
  • is the proof a simple for this statement? – Iconoclast Jun 26 '22 at 17:55
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    Well, any ball contained inside the square will not contain the entire square and then most (all but at most 4) of the points of the border of the ball will be in the square. The points of the border of the square can not be contained in any open set that is *disjoint* from the ball. – fleablood Jun 26 '22 at 20:21
  • " The points of the border of the square can not be contained in any open set that is disjoint from the ball" Put they can be in an open ball that is not disjoint from the original open ball. – fleablood Jun 26 '22 at 20:34

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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 $n\geq2$.

Ruy
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