Is the following statement true:
Any closed curve in $R^2$ enclose a region which has a nonempty interior.
If it is not true. Can you give an example of closed curve that has empty interior.
In this picture there are two curves pink and black. They are joined together. Is the union of these curves is a closed curve?
It really depends on what you mean with curve.
If you just mean some (nice) map $\gamma$ from the interval $[0,1]$ to $\Bbb R^2$ with the property that $\gamma(0)=\gamma(1)$, then the constant curve at a point certainly does enclose an area with empty interior. More interesting counterexamples are given by space filling curves.
If you assume that your map is injective, then indeed the Jordan curve theorem answers your question to the positive. While a seemingly obvious statement it is quite hard to actually prove.