Topology/Order

Recall that the usual topology $𝒰$ on $ℝ$ is defined so that the open intervals $(a,b)$ for $a,b\in ℝ$ form a base for $𝒰$. It turns out that this construction can be generalized to any totally ordered set $(X,\le )$.

Definition

Let $(X,\le )$ be a totally ordered set. The topology $𝒯$ on $X$ generated by sets of the form $(-\mathrm{\infty },a)$ or $(a,\mathrm{\infty })$ is called the order topology on $X$