Say we have a partially ordered set. What so you doubt? (1) The set of intervals $[x,\rightarrow[$ \left[x,\rightarrow\right[$is a base for a topology. (2) Any intersection of open sets is open. (3) The closure of$\{x\}$is$]\leftarrow,x]$. \left]\leftarrow,x\right]$. They all look OK to me...
Say we have a partially ordered set. What so you doubt? (1) The set of intervals $[x,\rightarrow[$ is a base for a topology. (2) Any intersection of open sets is open. (3) The closure of $\{x\}$ is $]\leftarrow,x]$. They all look OK to me...