If $C$ is a closed subset of a metric space, then $\bar{B}(C, r)$ is a closed subset also?

Question

Let $(X,d)$ be a metric space and $C$ be a closed subset of $X$. Let $\bar{B}(C, r) :=\{x\in X \ | \ \exists c \in C \ \textit{s.t.} \ d(x, c)≤r \}$. Then is $\bar{B}(C, r)$ a closed subset also?

Is the statement true if $X$ is a normed vector space over $\mathbb{R}$ and $C$ is a closed, bounded convex subset of $X$?

Answer 1

Oberve the space $X=\mathbb R-\{0\}$ equipped with subspace topology inherited from $\mathbb R$.

Then $C:=(0,\infty)$ is a closed subset.

But $\overline B(C,1)=(-1,\infty)\setminus\{0\}$ is not closed.