Topology/Order

Recall that a set $X$ is said to be totally ordered if there exists a relation $\leq$ satisfying for all $x,y,z\in X$

1. $(x\leq y)\land (y\leq x)\implies x=y$ (antisymmetry)
2. $(x\leq y)\land (y\leq z)\implies x\leq z$ (transitivity)
3. $(x\leq y)\lor (y\leq x)$ (totality)

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

Definition

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