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**Siron** Hi, I need some help with the following question:

Let $\displaystyle (X,d)$ be a metric space and $\displaystyle A \subseteq X$. Define the set $\displaystyle A^{(c)} \subseteq X$ for $\displaystyle \epsilon>0$ as $\displaystyle \{ x \in X | d(x,A) \leq \epsilon \}$

Prove the following statements:

(1) $\displaystyle \forall \epsilon \geq 0: \overline{A} \subseteq A^{(c)}$

(2) $\displaystyle \overline{A} = \bigcap_{\epsilon \geq 0} A^{(c)}$

(3) $\displaystyle A^{(c)} \ \mbox{is closed}$

(4) $\displaystyle \partial A^{(c)} \subseteq \{ x \in X | d(x,A)=\epsilon \}$

(5) Does there exist an example wherefore $\displaystyle \partial A^{(c)} \subset \{ x \in X | d(x,A)=\epsilon \}$

Proofs:

(1) Suppose $\displaystyle x \in \overline{A}$ thus by definition $\displaystyle d(x,A)=0 \leq \epsilon$ (with $\displaystyle \epsilon \geq 0$) therefore $\displaystyle x \in A^{(c)}$