Right ordered ray
As a set it is $\mathbb{R}_{\ge 0}$. The open set are intersections of intervals of the form $(a,+\infty)$ with $\mathbb{R}_{\ge 0}$
As a set it is $\mathbb{R}_{\ge 0}$. The open set are intersections of intervals of the form $(a,+\infty)$ with $\mathbb{R}_{\ge 0}$