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Suppose $X$ is a metric space and $S \subseteq X.$ Then, according to my textbook, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.

(Notations Used: $S^o$ refers to interior of $S$ . If $x \in X, dist(x,S) = \{\inf (d(x,s)~|~s \in S)\}$ )

I think that $S^o$ should be instead equal to $\{x \in S~|~dist(x,S^c)>0\}$.

$S^o$ is defined as $S - \partial S~~~~~..(1)$

where $\partial S = \{~x \in X ~|~dist(x,S)=0=dist(x,S^c)~\}~~~...(2)$

So, to obtain $S^o$, remove all those points from $S$ which satisfy $dist(x,S^c)=0$.

This yields $S^o=\{x \in S~|~dist(x,S^c)>0\}$.

Is it possible that my textbook has a possible error with respect to this definition?

Thank you very much for your help.

MathMan
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1 Answers1

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The textbook is (also) correct.

If $x\notin S$ then $\operatorname{dist}(x,S^c)\le d(x,x)=0$, hence $\{\,x\in X\mid \operatorname{dist}(x,S^c)>0\,\}\subseteq S$.

Hagen von Eitzen
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