Let $K$ be a compact subset of $\mathcal{R}^n$ with nonempty interior. Let $x_0\in \partial K$ (boundary of $K$). I want to show there exists a $\delta$ such that for any $\delta'<\delta$, $B_{\delta'}(x_0)\cap int\{K\}$ is connected. I don't even know if the statement is true. I asked a similar question (here Connectedness of intersection of two sets) but I did not formulate the question well so I am posting as a new question. @ECL said this is equivalent to asking if $K$ is locally connected. If I assume $K$ is locally connected, would the statement be true? If so, how can I make the argument rigorous? If not true, could you please provide a counter example?
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