Find the closure of A∩B I have  two sets, $A= \{1,2\}$ and $B=\{2,3\}$. $A \cap B = \{2 \}$. The definition of closure states, the closure of any set A is the smallest set that contains A. So I do not know how to use this to determine the closure of (A∩B)
According to your defnition the closure of a set $A$ $cl(A)$ is the smallest closed set $B$ which contains $A$.
Now a singleton set is closed. Therefore $\{2\}$ is smallest closed set which contains $\{2\}$.Hence $cl\{2\}=\{2\}$
Closure is the set $\overline{A}$ such that $\forall a\in \overline{A}$: $a\cap A$ is not empty. 

The closure of a single point would be that point itself, as it is the only point that always has a nonzero intersect with set described by that point!