$A$ is $(-1, -2) \cup (1, 2)$, and these are two disjoint sets whose union makes up $A$, so it fits the definition of disconnected but the book says that $A$ is a domain (it is open and connected). How is this set connected?
2026-04-02 01:31:38.1775093498
Why is $A = \{x \mid 1 < |x| < 2\}$ connected?
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The set $\{{\bf x} \in \Bbb R^2 \mid 1 < \|{\bf x}\| < 2\}$ is connected because it is path-connected. You can make a path between two points there using polar coordinates, for example.
The set $\{ x \in \Bbb R \mid 1 < |x| < 2\}$ is indeed $(-2,-1)\cup (1,2)$, so it is disconnected.