how can I show that the translate of an open set is open and that the translate of a closed set is closed Let $P$ be an open set of real numbers and let $C$ be an closed set of real numbers.  Let $x$ be a real number.  How can I show that $P + x$ is open and that $C + x$ is closed?  

It's been a while since I worked with topology and I'm sort of stuck.
Since $f:\mathbb{R}\to\mathbb{R} , f(u) =u+x$ is homeomorphism
Since $f:\mathbb{R}\to\mathbb{R} , f(u) =u+x$ is homeomorphism