Boundary of a general plane domain?

Question

Is it trivially true that the boundary $\partial U$ of a domain $U\subset \mathbb{R^2}$ is a finite (or countable?) union of disjoint Jordan curves?

Answer 1

No, it is not true (let alone trivially true). Consider $U = \mathbb{R}^2 \setminus \{(0,0)\}$. Then $\partial U = \{(0,0)\}$, which is not a union of Jordan curves (because a Jordan curve is always infinite...).

Answer 2

The boundary of the following shape is self-intersecting, hence no Jordan-curve.

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