So I have a problem. If we consider a metric space $(M,d)$ and a subset $K$. Is it true, that for $x\in M\setminus K$ we have that $$\inf\{d(x,y):y\in K\}=\inf\{d(x,y):y\in \overline{ K}\}=\inf\left\{d(x,y):y\in \partial K\right\}?$$ I am quite confident that is not true, but I'm not sure why. Can anyone provide counterexamples or a proof for each equation? Thank you so much!
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3The first equality is true (see for example here), but the second is not : for example, if $K$ is clopen, then $\partial K = \emptyset$ so the equality cannot happen. – TheSilverDoe Jan 04 '23 at 22:13
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allright, thanks:D – Adronic Jan 04 '23 at 23:20