Let $\Omega\subset\mathbb{R}^2$ be a bounded domain (open and connected). Assume that $$\operatorname {int}(\overline{\Omega})=\Omega\tag{1},$$

where "$\operatorname{int}$" denotes the "topological" interior of a set. Fix some $p\in\partial\Omega$. Can I find $p\neq q\in \partial\Omega$ and $h\in C([0,1],\partial\Omega)$ such that $h(0)=p$ and $h(1)=q$?

Hypothesis $(1)$ implies that every point in the boundary is not isolated (in the boundary). Its seems to me that this would implies what I want, but I can't find any argument. Any idea is appreciated


Not necessarily. For $n \in \mathbb{Z}^+$, let

$$U_n = \left(\frac{1}{n},\frac{1}{n} + \frac{1}{2^n}\right) \times \left(-\frac{1}{n},\frac{1}{n}\right),$$

and let $$Y = \overline{\bigcup_{n=1}^\infty U_n}.$$

Let $\Omega = B_5(0) \setminus Y$. Then $\Omega$ is a regular open set,

$$\overline{\Omega} = \overline{B_5(0)} \setminus \bigcup_{n=1}^\infty U_n,$$

and the path component of the point $(0,0) \in \partial \Omega$ in $\partial \Omega$ consists of only that point.


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