$x$ is normal space and we recognize him by his picture in $βX$. show that every $c_1 c_2$, close and disjoint sets in $x$ also the closure of $c_1$ and $c_2$ (in the closure of $x$) is disjoint.
i tried to bulid the Stone Čech compactification and see where it takes me but i didnt know what to do with the closure.
Use Urysohn's Lemma to get a continuous function to $[0,1]$ sending $c_1$ to $0$ and $c_2$ to $1$. That function extends continuously to $\beta X$.