# Why order topology is defined with open intervals, not closed ones?

In Wikipedia order topology is defined by the subbase consisting of $(a,\infty)$ and $(-\infty,b)$.

Why is not it defined by intervals $[a,\infty)$ and $(-\infty,b]$ instead? Is this Wikipedia definition (with open intervals) accepted by all or absolute most of mathematicians?

Consider an one-point ordered set (call this point $0$). With the Wikipedia definition we have an empty subbase. Isn't it better to have the subbase consisting of the set $\{0\}$?

If you define the intervals $[a,\infty)$ and $(-\infty,b]$ to be open, then you just get the discrete topology. Indeed, for any $a$ in your set, $\{a\}=[a,\infty)\cap(-\infty,a]$ would be open. On the other hand, defining it with open intervals gives the standard topology in the case of $\mathbb{R}$, which is pretty strong evidence that this definition is useful at least in some contexts.