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Let $Q$ be an open connected and bounded subset of $\mathbb{R}^2$. Is it true that for any two points $x,y\in \partial Q$, there is a path $\gamma \subset \overline{Q}$ which connects $x$ and $y$? I know that $\overline{Q}$ is connected and $Q$ is path-connected, but I still wasn't able to answer my question.

Mandelbrot
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    https://math.stackexchange.com/questions/286896/is-the-closure-of-path-connected-set-path-connected Does this solve your question?@Mandelbrot – pmun May 02 '21 at 13:45
  • Thank you, but I already saw this thread. I was hoping to get a positive answer since I only want to connect two points on the boundary and not arbitrary pairs of points. Also $\gamma$ isn't necesarrily going through any interior points. – Mandelbrot May 02 '21 at 13:50
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    Just take the given example and consider one boundary point on the vertical segment and one boundary point which is not. – Moishe Kohan May 02 '21 at 13:53

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