Epsilon-neighborhoods and epsilon-closures of a subset

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(a)(b)

Let $(X,d)$ be a metric space. For any subspace $A \subset X$ and real number $\varepsilon > 0$ , let

O_{\varepsilon }(A) = \left\{ x \in A : d(x,a) < \varepsilon \text{ for some } a \in A\right\} ,

C_{\varepsilon }(A) = \left\{ x \in A : d(x,a) \leq \varepsilon \text{ for some } a \in A\right\} .

(a)

Prove that $O_{\varepsilon }(A)$ is an open subspace of $X$ .

(b)

If $A$ is compact, show that $C_{\varepsilon }(A)$ is closed in $X$ . Must $C_{\varepsilon }(A)$ be closed for a general subspace $A$ of $X$ ?

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.