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Let $X$ be a path-connected metric space, and $U\subseteq X$ an open, non-empty, proper path-connected subspace.

There can exist points $x\in \partial U$ such that $U\cup \{x\}$ is not path-connected, and for spaces like $X = \mathbb{R}^n$, there always exists $x\in \partial U$ with $U\cup \{x\}$ path-connected.

Does there always exist $x\in\partial U$ with $U\cup\{x\}$ path-connected?

Jakobian
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    A remark since the question is already answered. There is a relevant notion in continuum theory: $x\in\overline{U}$ is accessible from $U$ if there is a path $p\subset U\cup{x}$ with $x\in P$. If $X$ is a Peano continuum and $U\subseteq X$ is open, then a dense subset of $\partial U$ is accessible from $U$ – Alessandro Codenotti Jun 10 '23 at 06:26

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Pick $x \in U$ and $y \in U^{c}$. Let $f: [0,1] \to X$ be a path connecting $x$ and $y$. Set $s=\inf f^{-1}(U^{c})$. Then $U\cup \{f(s)\}$ is path connected as we can connect $x$ with $f(s)$ through $f|_{[0,s]}$ and we have $f(s) \in \partial U$.

FZan
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