Here is the text of Exercise:

2 a) Let $X$ be an ordered set. Show that the set of intervals

$\left[x, \rightarrow\right[$ (resp. $\left]\leftarrow, x\right]$)

is a base of topology on $X$; this topology is called the right (resp. left) topology of $X$. In the right topology, any intersection of open sets is an open set, and the closure of $\{x\}$ is the interval $\left]\leftarrow, x\right] $.

The above one was from English edition. I translated French edition and found the same text.

Should not be $X$ a totally ordered set ? And is not that the set of intervals should be $\left]x, \rightarrow\right[$ in place of $\left[x, \rightarrow\right[$ ?

Is this an errata ?