For a subset $A$ of a metric space which of the following implies the other three

Question

For a subset $A$ of a metric space which of the following implies the other three ?

$a)$ $A$ is closed

$b)$ $A$ is bounded

$c)$Closure of $B$ is compact ,for every $B \subseteq A$

$d)$ $A$ is compact

I thinks option d) will correct .

is it true??

Answer 1

(d) is right, as a compact set is closed and bounded in any metric space. and also if $A$ is compact then for $B \subseteq A$, $\overline{B}$ is closed in $\overline{A}=A$ and hence compact too.