Let $S_{1} = \{ (x,y) \in \mathbb R^{2} | y \geq \frac{1}{x}, x> 0 \}$ and $S_{2} = \{ (x,y) \in \mathbb R^{2} | x = 0, y \leq 0 \}$. Now $S_{1} + S_{2} = \{ (x,y) \in \mathbb R^{2} | x > 0, y \in \mathbb R \}$, not clopen. Why is it not open? It certainly does not contain all boundary points such as $(0,0)$ but for open, every point requires a neighborhood (in Euclidean space, more here).
What is the $S_{1} + S_{2}$? It is not union but someting else? What about $S_{1} \cup S_{2}$? What is it like? It does not contain the boundary points so it is not closed. But I am now uncertain because the $S_{1}+S_{1}$ is not closed. My intuition falls here short.