Q:

Is closure of a compact set itself? 

I came across this question: 
If a point $a$ and a compact set $B$ have $d(a,B)=\inf\{d(a,b):d\in B\}=0$, does it mean $a$ is in the closure of $B$?  If it is, can we say since $B$ is compact, its closure is itself so $a$ is also contained in $B$? 
Any help is appreciated. 

A:

In a metric space(more generally in a Hausdorff topological space) any compact set is closed.  Hence a compact set is its own closure.