Define a topology on $\displaystyle \Re$(by listing the open sets within it) that contains the open set (0,2) and (1, 3) and that contains as few open sets as possible.
You basically need to go back to the definition of a topology.
If $\displaystyle \mathcal{T}$ is a topology on the set $\displaystyle X$,
then $\displaystyle X$ and $\displaystyle \emptyset$ are in $\displaystyle \mathcal{T}$
And if there are A, B are open in $\displaystyle \mathcal{T}$, then $\displaystyle A \cap B$ is open in $\displaystyle \mathcal{T}$ also.
And if there are A, B, ... are open in $\displaystyle \mathcal{T}$, then $\displaystyle A \cup B$ is open in $\displaystyle \mathcal{T}$ also.
Can you take it from here?
It's also important to know: If each set $\displaystyle \mathcal{A}_1,\mathcal{A}_2,\cdots,\mathcal{A}_n$ is open, then
$\displaystyle \mathcal{M}_1:=\bigcup_{k=1}^{n}\mathcal{A}_k$
$\displaystyle \mathcal{M}_2:=\bigcap_{k=1}^{n}\mathcal{A}_k$
are open, if closed then closed. There's also a simple proof.