If G = \left\{ \left(x, 1/x\right) : 0 \neq x \in \R \right\} is the graph of f(x) = \frac{1}{x} and if and Y = \{ 0 \} \times \R is the y-axis in X = \R^2 then the Minkowski sum of these two closed subsets of the plane is the open set G + Y = \{ (x, y) \in \R^2 : x \neq 0 \} = \R^2 \setminus Y consisting of everything other than the y-axis. This shows that the Minkowski sum of two closed sets is not necessarily a closed set. However, the Minkowski sum of two closed subsets will be a closed subset if at least one of these sets is also a compact subset.
